Tamable Mobs and how to Tame Them - Part I
It's possible to gain benefits from the tamed mobs, like with dogs, but in the cases of parrots they don't provide any in-game benefit apart from looking aesthetic.
So let's begin with perhaps the most well-know tamable mob in the whole of Minecraft - Wolves!
Wolves spawn most often in Taiga biomes. They are neutral mobs which mean that as long as you don't hit them, they won't try and attack you. Be wary though, if you do attack one, then all the wolves in a nearby area will become hostile toward you, so you may easily be overcome by them.
Approach the wolf you want to tame, holding bones in your hand. It usually won't take more than 4 bones to tame a wolf, but it can theoretically require any number of bones as each bone has a third chance of taming the wolf.
Right click the wolf with the bone. Keep doing this until love-hearts appear around the wolf. It will also sit down and gain a red collar. Congratulations! You have now turned the wolf into your dog!
You can change the colour of the dog's collar by right-clicking the dog, ensuring you have the dye held in your hand. You can right-click the dog with nothing in your hand in order to get your dog to sit down/stand up. As you walk, it will follow you wherever you go and it will attack anything you attack.
Parrots can be found in the Jungle biome in 5 brilliant colours. You need to find a parrot and right-click it, while holding seeds in your hands. Again, love hearts will appear around the parrot. As with dogs, it can be made to sit down or get up by right-clicking it with an empty hand. If you move closer to the parrot, it will probably perch on your shoulder. You can switch views by clicking F5 in order to see this clearer.
If you have your volume turned up, then the parrot may warn you of nearby hostile mobs by imitating their noise. If you play music using a jukebox and take your parrot with you, then the parrot will begin to change colour and start dancing.
Next up, let's take a look at ocelots!
Similar to parrots they also spawn in the jungle biome, but their behaviour is far more complicated.
The taming process is quite complicated, so heed the instructions carefully.
You will need to be holding raw fish, it doesn't matter what kind but just ensure the fish isn't cooked. Hold the raw fish and stand by the ocelot. You can't stand too close (within 6 blocks and the ocelot will dash) but you can't stand too far (further than blocks away and the ocelot won't recognise you standing there). The ocelot will start peering at you with curiosity, more interested in the fish than in you. Don't make any sudden movements otherwise it's likely that the ocelot my get scared and flee. The ocelot will walk up to you and when it is within 3 blocks you should right-click it holding the fish, thus feeding it the fish. One fish should probably do the trick, but some times you may need to feed it more times.
One the ocelot is tamed, it will change colour and is now your cat. You can make it sit, get back up and it will automatically follow you.
The main benefit with having a pet cat is that they will deter Creepers from approaching you, and so keep them at a distance. Beware though, for all the benefits of having cats...they sometimes sit on top of your chests :P