He enables the combo, as well as brings an often giant Construct token onto the battlefield with him, and acts as both a mana and card advantage engine. The token is a legitimate threat on it's own given the number of artifacts we run, as well as providing a beefy blocker. Between Urza's 4 toughness and the Construct token, Urza provides a lot of defense against aggro strategies all on his own. His second key ability is his ability to tap any artifact you control for blue mana. Importantly this still works under Stony Silence because it's an ability of Urza, not the artifacts, and it also operates while Urza is being named by Pithing Needle since it is also a Mana Ability. This allows Urza to generate a ton of mana fast, which leads right into his final ability. His final ability is likewise great and allows Urza to effectively draw many extra card every turn even without the combo. Always play 4 copies.
Thopter Foundry - Probably the second best card in the deck. Enables the combo and can also be a relevant threat on it's own. Even without Sword of the Meek, sacrificing artifacts to gain life and make flying 1/1s can provide a viable clock in longer matches, especially against control. Foundry is often your preferred turn 2 play most of the time, and you should probably play 3-4 copies
Sword of the Meek - Easily one of the worst cards in the deck in a vacuum, it is necessary for the combo. It does very little on it's own, but it's important to remember it does have an equip cost that you can pay to pump up a construct, or an Urza, even if you'll very rarely actually use it. Really nice that this generally does not even need to resolve, as if it is countered and ends up into the graveyard, the combo can begin just as easily by sacrificing another artifact to make a token and return the sword. Only 1 or 2 copies are usually required in the deck.
Mox Opal - Obvious inclusion in any artifact based deck to speed up the deck. Our deck is not so heavy on cheap artifact that you can expect to be casting 2 drops on turn 1 very often, as the Opal is often not turned on until turns 2 or 3, however it's still a very good card for the deck. You should probably play 4 copies, or 3 at the minimum.
Mishra's Bauble - 0 drop card that cantrips (though delayed by a turn), which helps enable Mox Opal, Whir of Invention, etc. You can also bauble yourself with a fetchland in play, then based on if you want the top card or not, shuffle it away with the fetchland before your next draw. When to sacrifice Bauble and when to keep it in play for Opal or Whir is rarely a clearcut answer. You need to analyze your hand, and your opponent to determine if the extra card on your following turn is worth it, or if it's better to keep it in play to either enable an extra improvise on Whir of Invention, or if it's needed for Mox Opal's metalcraft. Play 4 copies
Arcum's Astrolabe - Astrolabe is a card that is easily overlooked or dismissed, however is it has become a major core of the deck since it's printing. Astrolabe is the only 1 cmc artifact in modern than can draw a card when it enters, which alone pretty much makes it worth an inclusion, however the additional ability to filter mana to any colour without having to sacrifice itself makes it excellent at fixing mana, enabling whir, or generating additional mana with Urza. The only downside is that it requires a very snow basic heavy manabase to enable it on turn 1 consistently, however given the nature of the colour requirements in the deck, this is generally very easy to justify. Play 4 copies
Whir of Invention - Whir enables the toolbox of the deck, as well as acts as an instant speed tutor for our combo pieces of Foundry and Sword. The fact that you can't reduce the coloured mana cost does make it a little slow at times, but still well worth it. Whir's triple blue cost also means you have very little room for lands that don't produce blue mana in the deck. Generally you will want 2-4 copies.
There are several options for colour choices with the deck, but given the blue cost of Urza, Whir, and Thopter Foundry, blue will always form the core of the manabase. The hybrid cost of Thopter Foundry will also force you into black or white as a secondary colour. Occassionally you will end at a 2 colour deck, however the printing of Goblin Engineer that can put a Sword of the Meek into the graveyard for you, or act as a value engine, has often made red a key part of the archetype. This means the most common colour combinations for the deck will typical be Grixis or Jeskai, however straight Dimir, Esper, or even 4 colour decks are possible. Arcum's Astrolabe and Mox Opal do an excellent job of fixing whatever colours may be needed. Green is splashed the least often of all the colours, as it's only real options tend be more reactive cards like Abrupt Decay or Assassin's Trophy, it however does remain a consideration.
Since the Urza + Foundry + Sword combo doesn't actually let you win the turn you go off, but the infinite mana lets you dig through your whole deck, most people choose to play a single copy of a card that lets them win on the spot. Here I'll outline the most common options and their pros and cons.
Spine of Ish Sah - Doesn't actually win on the spot, but you can sac it to Foundry, putting it back into hand, and using your "infinite" blue mana to recast it, repeating until you've blown up every permenant they have, including lands. I can't think of any decks that would still be able to win from that spot. It's also able to be fetched with Whir of Invention, and can potentially answer a single hard hate card like Leyline of the Void, or Stony Silence. Downsides are that it is expensive, so casting it without an Urza can be tough, it also doesn't work against multiple hate cards.
Ghirapur Aether Grid - Great at pinging off small creatures, pressuring walkers, and acting as a wincon against even a Stony Silence. Does require red mana, and can be shut off with Pithing Needle effects, also can't be Whir'd for.
Scrap Trawler - Works with Pyrite Spellbomb to burn them out. Sac the spellbomb for 2 damage, then sac a Mox Opal to make a thoper, then sac sword to get back Spellbomb, and then sac sword again to get back Opal to pay the red to activate spellbomb again. Also can be tutor'd with Whir, and works as a great value engine when sacrificing artifacts for value with engineer or foundry. Scrap Trawler additionally has an infinite combo without Urza. If you have Scrap Trawler, Thopter Foundry, Sword of the Meek, and 2 Mox Opals, you can pay Foundry's activation cost with Mox Opal, then Sacrifice Sword, returning the 2nd Mox Opal from the graveyard with the Scrap Trawler trigger, as well as returning the sword. Repeat as many times as you wish as each time you return 1 Opal, the other will die due to the Legendary Rule. Weak to creature removal and graveyard hate.
Grinding Station - Mills the opponent out, since each Foundry activation yields 2 triggers (for sword and the token) and you can sac the token each time to mill your opponent. Also works as a small mana engine with Urza, allowing you to tap it for mana, then untap it if you cast an artifact, tapping it again for mana. Interestingly also triggers off opponent's artifacts entering, so possible extra utility in the mirror
Time Sieve - Can allow you to take infinite turns. Also works as an infinite combo without Urza, with just Thopter Foundry, Sword of the Meek, Time Sieve, and at least 5 mana. Allowing you to make 5 tokens, sac them to take an extra turn, and eventually net more and more leftover tokens as you play more lands.
Mirrodin Beseiged - Mirrodin option can make tokens for value, while the Phyrexia option can win the game for you on your end step. Not vulnerable to artifact hate, however the Mirrodin side is worse than Sai, Master Thopterist and the Phyrexian side is poor against graveyard hate.
Here I'll outline other commonly played or considered cards and group them by colour. I will include multicolour cards that are partly blue with the card of the 2nd colour (ie: UW cards with the White cards)
Grafdigger's Cage - Commonly played maindeck graveyard hate. 1 mana artifacts are good, graveyard hate is good in this graveyard heavy format. Usually only play 1 max in the main
Nihil Spellbomb - Similar to Cage above. One shot effect, but can also cycle in matchups where the graveyard hate is relevant, so it's a decent maindeck card.
Tormod's Crypt - Can't cycle when it's bad, but costs 0.
Pithing Needle - Also a commonly played 1 or 2 of in the main. There are few matchups where it's dead, and it's great against planeswalkers.
Pyrite Spellbomb - Artifact based removal spell that can also cycle when not needed. Pretty solid option overall
Ichor Wellspring - Cantripping artifact. Works great with Goblin Engineer or even a nice option to sac to Thopter Foundry. Most lists only run 1 for when Engineer needs to grab a value option in the face of potential graveyard hate, but anywhere from 1 to 4 copies are seen in various lists.
Mind Stone - Helps you ramp into Urza, while also cycling for a card late game
Pentad Prism - Also helps you ramp into Urza. Slightly faster than Mind Stone, at the cost of being a terrible top deck late game
Ensnaring Bridge - Great against a lot of creature decks. Some decks can't beat it in game 1. 1 copy is the most commonly played number, as the effect isn't needed as much with the combo, but still often an excellent option to have access to in a deck with Whir
Chromatic Star - Fixes colours, can cantrip. Downside is that it only cantrips when you sac it, which means Leyline of the Void/RiP shut it off, and it's not reusable. Definitely much worse than Arcum's Astrolabe, but it's another option for a 1 mana artifact that can draw you a card
Terrarion - Same deal as star, but entering tapped and costing 2 to activate makes it a worse version of star.
Sai, Master Thopterist - 1/4 blocker than makes flying tokens and can help draw you extra cards. It's extra silly with Urza as the new thopters will net mana. Sacrificing a Sword of the Meek to his ability is often good, as if you draw and cast an artifact, it will make a token, returning Sword to the field. Excels against midrange/control decks and can win games almost by himself. He also makes it sometimes worthwhile to sandbag 0 mana artifact in hand until you've cast a Sai, if you don't need them sooner, so you can get immediate value.
Ceremonious Rejection - Sideboard card that is fairly commonly played as primarily an answer to Karn, the Great Creator
Counterspells - Always an option, though don't always play nicely with cards like ensnaring bridge, or the more tapout nature of the deck.
Tezzeret, Agent of Bolas - Probably one of the best alternate wincons. His +1 can dig you for answer, as well as quickly threaten a lethal ultimate. His -1 can also put a lot of pressure on the opponent quickly, as targeting an artifact that doesn't have summoning sickness will effectively give you a hasty 5/5.
Dead of Winter - Really effective wrath effect that is enabled by our manabase and Astrolabes
Fatal Push - Solid removal, is there more to say?
Discard - Effective vs control and combo decks
Leyline of the Void - While technically playable without black, Leyline always feels terrible if you can't cast a topdecked one late, even if at that point its effectiveness is lower. Possibly the most effective, but narrow, graveyard hate card we have.
Collective Brutality - Discard, removal, and damage all in one. Works well with Ensnaring Bridge, and amazing against Burn.
Battle at the Bridge - Solid removal that can hit even large creatures and indestructible creatures. Incidently lifegain is also often relevant, especially against burn. Main downside is sorcery speed, and that it is bad against Gaddock Teeg
Teferi, Time Raveler - Excellent planeswalker. Can protect your combo while also dealing with hate cards and providing card advantage with his -3.
Wear // Tear - Obviously need to also be in red, but great removal against opposing hate cards, which are often enchantments or artifacts.
Deputy of Detention - Creature based removal for hate cards. Being a creature it can often avoid the Negate type effects people play, or the artifact/enchantment removal, however is vulnerable to creature removal. Also not exiling your own cards can be helpful in the mirror
Detention Sphere - Similar to above, but vulnerable to different removal
Tempest of Light - For when you really just hate enchantments
Lightning Bolt - Solid removal spell that can hit players and pressure walkers
Galvanic Blast - Same as Bolt, however with metalcraft it's slightly better than bolt, and slightly worse without metalcraft. Blast is more commonly played than bolt, but Bolt is more consistent if you're worried you may not always have metalcraft. Especially against Infect, casting Blast with metalcraft is not super reliable on turn 1, and a Mutagenic Growth will make you wish it was a bolt.
Magmatic Sinkhole - Can sometimes be better than even galvanic blast, as our heavy fetchlands fuel delve, but probably can't afford more than 1 in the 75.
Assassin's Trophy - Similar to Decay, but counterable against UW, but hits Leyline of the Void.
Maelstrom Pulse - Hits everything, includes multiple copies, but 3 mana and sorcery speed.
Back to Nature - For when you really just hate enchantments
Other Spells (3)