How do I hire companions in Skyrim?

I was playing a sneak style character with companion and liked that, but in one battle with a dragon my companion died. Can I hire a new companion in Skyrim? If yes, then where to hire them?

Solution 1:

You can find them in several Inns and some are quest-related. Here is a incomplete list of several followers.

Additionally you can buy a wardog for 500 gold in front of the stables in Markarth who is independent of a human follower (you can use both).

Solution 2:

Companions are people you work with usually, or sometimes affected to your service.

For example, Lydia is affected to you if you became Thane of Whiterun. There is a mage that will accept to follow you in the Winterhold college if you help her.

I guess there is more, but I don't know all of them.