A fursona is the animal version of yourself and most of them are animals with fur (which is why they aren't feathersonas, or scalesonas) which are mostly dogs and wolves, but sometimes cats, lions, hyenas, foxes, horses.

Your fursona is like an animal alter-ego, the people who make them are known as "furries" look them up in wikifur. They tend to be drawn in Lion King/Balto style.
As for avatars, unless you're referring to Avatar's Na'vis then that would be your icon the little picture in the corner with your name underneath, I think mine's a *super old* picture of a brown canine or something ^^'

Your fursona is another form of you that you create. It can be male, female, animal, look however you like and might even have a similar history to yourself.

An avatar is the profile picture you use, but if this is in creating a character, then it would be the face shot of your character/fursona.

Your fursona is what represents /you/. It's different from the rest of your characters.