Attention:
If you want to actually learn to play Nasus, not just read how to build him, go read Hyfe's guide on SoloMid:
http://solomid.net/guides.php?g=151
Anything I could write would be either plagiarism or inferior.

Nasus is a bruiser (tanky dps) champion, but he leans more towards being a full tank. The reason for this is that his damage scales relatively poorly with damage items, so he just builds tank items. Nasus's main strength is his absolute dominance in the very late game; he is very hard to take down and can Siphoning Strike squishies for over half their health. His main weakness, however, is being outranged with no initiation or charge. This makes Corki and Ashe very powerful against him, which is why he is considered less than great (These are currently the two best AD ranged carries). He is generally a good pick in a ranked game when neither of these two are on the other team, and you have your main tank in the jungle. He can be picked safely in blind pick games. He generally can take solo top, but is just fine in a duo lane. Don't go mid. Although he can survive mid, send your carry there; they actually need the levels.

P. S.: Nasus is not a stand-alone tank.

Ghost: Ghost is an important spell on Nasus; really, there's no substitute for this one. Chasing, escaping, this one does it all.
Exhaust: I personally like Exhaust because you can do Wither Exhaust Wither for a sustained slow and a solid gank. This one can be substituted with Cleanse, Flash, or Teleport based on preference. I guess you should pick Smite if you plan on jungling, although there are so many better junglers.

Offense: Really, there isn't much you want in this tree. The only useful ones are Cripple, Sorcery, and Sunder , but it really isn't worth going very deep. Just take Cripple if you have exhaust.
Defense: The tree you want, it has (surprise) masteries that improve your defense. Ardor isn't great because neither stat does much for you. Take Willpower if you take Cleanse.
Utility: Throw your remaining points in this tree. You can definitely do 0/9/21 if you please, taking Nimbleness in defense. Overall defense tree is going to help more.

The two main alternatives to 1/21/8 are 0/9/21 (grabbing Nimbleness ) and 1/16/13 or 1/18/11 (grabbing Meditation ).

Fury of the Sands> Siphoning Strike> Spirit Fire> Wither

Always level up your ultimate. Take Siphoning Strike over Spirit Fire since it does more damage and excessively using Spirit Fire will absolutely drain your mana. Take one point in Wither at level 4 to help chase, but you don't need more than that.

Note: The skill sequence is reasonably flexible; the above skill sequence is just typical. More points in Spirit Fire would be good if you are getting pushed hard and more points in Wither is good when your team is ganking your lane a bunch. Just be careful because all of your Wither ranks can be ruined by Mercury's Treads.

Greater Mark of Desolation: Best for making Siphon Strike hit harder; enough said.
Greater Seal of Evasion: I really love the dodge for Nimbleness procs and the damage reduction.
Greater Glyph of Scaling Mana Regeneration: You need mana regen from some runes, and these aren't much worse than Greater Seal of Scaling Mana Regeneration. You can also take Greater Glyph of Cooldown Reduction, Greater Glyph of Scaling Cooldown Reduction, or Greater Glyph of Scaling Magic Resist, but then you have to take Greater Seal of Scaling Mana Regeneration. If you don't have any mana regen runes, you will have to buy Chalice of Harmony at some point in your build. Overall I think the dodge runes help more than CDR or MR.
Greater Quintessence of Movement Speed: Movement speed really helps Nasus (as you will see in the item purchases) so I think these are the best. You can also take Greater Quintessence of Health for more early game survivability, or Greater Quintessence of Desolation for more damage.

DISCLAIMER: YOU DO NOT BUILD THE SAME ITEMS IN THE SAME ORDER EVERY GAME. EVERY TEAM HAS DIFFERENT NEEDS, AND EVERY ENEMY TEAM HAS DIFFERENT STRENGTHS. THE BUILD AT THE TOP IS SIMPLY THE MOST COMMON BUILD.

In general, get plenty of movement speed on Nasus, he needs it to get into the fight.

Main Build:
Doran's Shield: Best starting item for Nasus. Health keeps you alive and health regen helps when you can't get hits to utilize Soul Eater.
Sheen: Try to refrain from returning to base until you can get enough for this and Boots. This makes you hit a lot harder with Siphon Strike and really buffs your early game damage.
Mercury's Treads: Keeps the other team from keeping you away; generally these are the boots of choice.
Aegis of the Legion: Great item in general, Nasus can pick it up easily since he is not too dependent on getting items for himself. This item can easily be substituted with Heart of Gold or Catalyst the Protector, or even The Brutalizer if someone on your team is already building Aegis and you plan on building Randuin's Omen, Banshee's Veil, or Youmuu's Ghostblade respectively.
Force of Nature: Everything about this item is great, plus movement speed is awesome on Nasus.
Frozen Heart: CDR lets you use Siphon Strike more, plus the aura helps out teammates.
Trinity Force: Upgrade your Sheen into this at some point. Buy Zeal if you are going to build one part before completing it. MORE MOVEMENT SPEED OM NOM NOM NOM.
Zeke's Harbinger: Great support item for your AD friends, while also boosting your own damage. Great for the last slot. Replace this with Youmuu's Ghostblade if someone on your team is already building Stark's or if your team has no other AD.

With these items you approach 500 movement speed on Nasus, which is absolutely critical to your success.

Other Good Items That You Might Use At Some Point:
Chalice of Harmony: As mentioned earlier, you need this if you don't have mana regen runes.
Boots of Swiftness, Ninja Tabi: Different teams mean different boots. If they have a lot of AD and little CC, take Ninja Tabi (helps Nimbleness procs too!). Take Boots of Swiftness against a team that is well rounded and has little CC. I could even see Ionian Boots of Lucidity for faster Siphoning Strike.
Banshee's Veil: Definitely outclassed by Force of Nature, but can help in certain comps. You can also take it over Frozen Heart if they are heavy magic.
Quicksilver Sash: If suppressions are killing you, pick this one up.
Randuin's Omen: A really solid and well-rounded armor item that I used to use before it got nerfed. It's still usable though. Take it over Force of Nature against mostly AD teams or in place of Frozen Heart if you are more concerned about your own durability. It's also good if you have to build Heart of Gold when you can't get any farm, but no last hits for Nasus pretty much means you are going to be underwhelming late game.
Guardian Angel: Not a great item for Nasus, but if you are getting focused hard and killed in team fights, it can definitely help reduce kill count for the other team. Plus the defense stats are good.
Thornmail: If they have an AD that is just mauling your health, pick this up and shut them down.
Youmuu's Ghostblade: Take this instead of Zeke's Harbinger if you have no other AD on your team or are carrying your team.
Frozen Mallet: Another alternative to Zeke's Harbinger. It gives more health but less damage and support. It also has some synergy with Force of Nature. I honestly haven't decided which one I like the best of these three.
Warmog's Armor: Take this instead of a damage item if you want to be hilariously durable, as long as they don't have any percentage health moves (Kog'Maw *shudder*). This also has good synergy with Force of Nature.
Last Whisper: May be useful if your team has too much AD and they are stacking armor.

Items That Seem Decent At First But Actually Aren't So Great:
Lich Bane + AP items: Don't build AP Nasus; it's not viable.
Bloodthirster, Infinity Edge, Phantom Dancer, etc.: These are items for AD carries and honestly don't increase your damage for as much as they cost.
Tiamat: You farm like a boss, and this doesn't stack up Siphoning Strike faster.
Sunfire Aegis: This is for tanks that have little damage output. Nasus does plenty of damage.
Spirit Visage: Seems like it might be alright but you don't really lifesteal enough to take advantage.
Wriggle's Lantern: The popular item of today. Lifesteal doesn't help that much; you already have that. Attack damage doesn't help Nasus as much as it helps others. It doesn't have that much armor. Minion proc doesn't really help at all. Wards aren't that expensive. This would be a very good item if you want to jungle but Nasus does not belong in the jungle in my opinion.
Madred's Bloodrazor: Attack speed isn't really Nasus's thing, so on-hit-effect items aren't so great on him.
Manamune: Runes should cover your mana and attack damage isn't that awesome for Nasus.
Shurelya's Battlesong: This one was more of a random thought; tell your support to get it and get yourself a Youmuu's Ghostblade.
Leviathan, Sword of the Occult: I am never fond of snowball items, especially on a champion that tends to have a relatively low kill count early game.

Undecided:
Atma's Impaler
Hexdrinker

Since I might get burned (haha funny) for saying that Sunfire Aegis is bad for Nasus (if anyone actually reads this guide...), I added a chapter just for it. Although it is a solid and well rounded tank item, it is not the best for Nasus (think Frozen Heart).

In general, the two main tank items that give both armor and damage are Sunfire Aegis and Frozen Heart. When considering buying Sunfire Aegis, you must ponder if it will increase your damage output more than Frozen Heart.

First I will list champions that can utilize it well and why:
Amumu: The AoE damage overlaps well with the mechanics of Despair and Tantrum, Curse of the Sad Mummy can keep enemies in range of the damage, and CDR from Frozen Heart does not affect Despair and Tantrum has CDR already from getting hit.
Galio: Sunfire Aegis increases his survivability, and melee champions will be burning in it while taunted from his ultimate. It's pretty good on Galio. Frozen Heart would work decently on Galio as well
Malphite: The health makes his Granite Shield bigger and the armor scales with Brutal Strikes and Ground Slam. Since he doesn't really do much damage, running around with a Sunfire Aegis on is a good option.
Rammus: The armor scales well with Spiked Shell and Defensive Ball Curl. Also Puncturing Taunt helps keep enemies in the AoE damage.
Shen: Sunfire Aegis is an insanely good item on Shen. The health increases the damage from Ki Strike, the armor makes Feint last longer, and Shadow Dash can make multiple people eat the AoE. Frozen Heart is really bad on Shen because CDR and mana don't help much when you have energy.
Singed: Sunfire Aegis helps out Singed because he can't really do much damage once he poisons everyone, so he ends up just running around doing AoE damage.

Garen, Mordekaiser, Dr. Mundo: Big meatballs that don't do much damage so they pick up a Sunfire Aegis, stacking well with their Judgment, Creeping Death, and Heart Zapper respectively. Don't pick these champions unless you like losing.

Looking at the previous champions, you can notice certain traits that make Sunfire Aegis better than Frozen Heart:
1. Damage that scales based on health and/or armor.
2. Not having mana.
3. Damage that scales poorly with CDR.
4. An ability that helps keep (multiple) enemies within the damage radius.
5. Another AoE radius ability that can stack well with Sunfire Cape.

Here's how Nasus's traits look with corresponding numbers:
1. Nasus's damage does not scale with health or armor.
2. Nasus does have mana.
3. Siphoning Strike damage scales great with CDR.
4. Nasus has only one CC, Wither, and it isn't going to help much with AoE damage.
5. This one is a bit tricky. This is the reason people think Sunfire Aegis would be good on Nasus: Fury of the Sands. Nasus can stand next to people and burn them with both his Cape and his Ultimate. However, a damage breakdown of Nasus would show that about half of his damage comes just from Siphoning Strike, increasing throughout the game. AoE damage from his Ultimate is no more than a quarter of his damage output, probably even less.

CONCLUSION: The CDR from Frozen Heart increases Nasus's damage more than the AoE from Sunfire Aegis, while also providing more armor and an aura for allies. This damage discrepancy arises because Nasus's main source of damage is Siphoning Strike, not Fury of the Sands.

Hope that clears things up for anyone who was wondering what armor item to build.