I think that the most useful Vanilla mob is actually the Mooshroom, although they are extremely rare and only found in Mooshroom Biomes. You can milk them infinitely for Mushroom Stew with only the need of a wooden bowl which is easily crafted, and drop leather, from which armour is made from, food (I think milking them is better though), and also can be milked for milk and sheared (emergencies) for mushrooms. :Creeper:
Probably the most useful Modded mob is the King from Orespawn. If you're playing in a modpack, the King has a chance to drop creative items or hard/almost impossible to get items. I know that you can get a Creative Strongbox from killing one. Of course though, they are really difficult to kill. A couple of Spiral Lucky Block's Really Unlucky Potions could bring it to half health, but... wait, I don't want this to become a Tips on Killing the King Post. You get my point.
As far as I'm concerned, in vanilla minecraft...it's got to be horses.
Cows, well I can eat pigs, chickens, fish, vegetables. Leather for armor? I can find some iron and make better on my own.
Any other food animal is the same way...you can find plenty else to eat.
Skeletons, yeah, they're valuable but even without bones/bonemeal, I can eventually get stuff to grow. And I can make a bow on my own.
However, with a good horse with speed, health and good jumping ability, I can go pretty much anywhere and get there much faster than on my own. I can also ride away from danger quickly if I'd like. You can hop on a horse, ride all day, go thousands of blocks and don't have to eat every couple minutes or get fatigued. The only other strictly survival mode of transportation is minecarts which require lots of layout, resources, time and only go whatever direction you laid it out to go. (I guess there's boats too but they're limited to overworld and water...). And even minecarts are slower than even slow horses.
Sheep (They are one of the only two ways via which we can obtain wool; the other being spiders. Drop meat as of 1.8), Cows (can be milked, and drop one of the best foods in the game and leather when killed), Skellies (drop bones which can be turned into bonemeal which can be used in expediting the growth of plants; also drop arrows and sometimes their weapon; may even drop armor)
Definitely Villagers. I always love coming across one of their villages, which make very nice outposts for those long adventures away from home. The trading system is also incredibly nifty and is how I get most of my Enchanted Books and other supplies I might need while out exploring.
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Friendliest wolfshark thing you'll meet. Maybe.
'Tis better to die for the Emperor than to live for yourself!