Alistar is a supportive tank that is mostly found in one of these two roles. He's a strong engager with a bunch of cc. In most roles movement speed is a key aspect of your item build, runes and masteries.
Most of his rune builds, masteries and items are basically the same in the builds but I put them all in to show the differences between those builds.
As I personally play Alistar mostly as support this is the build i did most detailled especially because a lot of thing do repeat and show up in his other builds too.

Strenghts:
- naturally tanky
- lot of cc
- good sustain (heal)
- strong ability combinations
- good engager
- one of the best protective supports


Weaknesses:
- Mana hungry
- requires good timing
- no ranged poke
- low damage output (unless AP) -> bad in 1 vs 1
- vulnerable in early game
- very dependent on his team
- long cooldowns

Very Important:

Movement speed: Speed helps you to successfully engage on enemy players and is so one of the most important thing to build on.


Armor: As you'll be playing on bot lane or jungle armor will give you durability against most damage you recive. Runic Bulwark and Frozen Heart are great choises here.


Health: Helps you survive in lane and is especially when dealing with magical and true damage.
A lot of your core items like Ruby Sightstone and you don't need to build pure health items like Warmog's Armor in most cases.

Important:

Mana Reg: As you're very mana hungry you'll need some mana reg in each of your builds. Philosopher's Stone and Shard of True Ice are great choises.


Health reg: Helps you to sustain in a lane when you don't necessarily need to use your Triumphant Roar or want to save mana. Included in some core items.


Cooldown reduction: As you have long cooldowns at least some of your items should be build to encounter that. Most of them are gonna be tanky items like Shurelya's Battlesong and Frozen Heart.

Less important:

Ability power: Ability power increases the damage infected by your cc abilities and your passive. It also increases the heal by your Triumphant Roar. Unless you play ap Alistar you won't build especially ap based items.


Magic resist: Unless you have to deal with 3 to 5 ap champ in enemy team Runic Bulwark and maybe Mercury's Treads should be the only mr items in your build as health helps you surviving magical damage as well and is a more general defense.

As a supporter Alistar gives great sustain to his lane mate and can be played with protection dependent ad carries like Kog'Maw. His cc is a great way to either initiate or escape while his heal gives you the time to wait for the right moment. As in jungle movement speed is the key to successful engages. Alistar can be played on a burst or sustain bot lane but is the completely wrong decision for a harass bot. As support max the heal first and use his cc abilities for engages or escapes.
Fulltank Ali is about the same purposes as support Ali but more based on strong engagements with his lane mate. That version is more about destroying the enemys' bot lane and so you need a lot of tankyness for you engages. The cooldown reduction on summoner spells makes sure your Flash will be ready in time so you can engage with Flash + Headbutt to knock the enemy ad carry backwards and deny his escape with Pulverize. You can use this combination as support as well.

When jungling your main intention will be to gank and so increase the lane pressure to your enemies. I recommend a supportive focussed build with a little more movement speed than as supporter. That's why Trinity Force is more recommended than Iceborn Gauntlet. An early Shurelya's Battlesong is really strong for ganks. Alistar is much slower in farming the jungle than bruiser junglers like Rengar, carry junglers like Master Yi or mage junglers like Elise or Diana. In order to equalise that you need to help you team with strong ganks using your cc.

Though it is possible to play ap Alistar i do not recommend to do so because it's very hard to play. If you decide to do so the perfect timing of your skills will help you to be sucessful. Your intention will be to Headbutt and Pulverize your enemy mid champ in one step. If you suceed doing so you deal a lot of damage to your enemy if not you don't have a good chance to engage on your enemy without the help of your team mates or taking a lot of damage. Farming is really hard with ap Alistar as you need to be at least in front of the enemys' minions to Q or autoattack them.
It may be quite funny in bot games but better don't try that for normal or ranked games.