Nevertheless is used when what is expected is contrary to what happened. For example, "I did something unkind to you but, you were kind to me nevertheless." You would've expect them to be unkind to you because you did something bad but, they were still kind to you. Which is contrary to what you would have expected. However, do not get nevertheless and although mixed up! Nevertheless is more properly used at the end of a sentence while although is used at the beginning.
However is used when you are explaining an exception or justification. This is identical/interchangeable to but. Although, you must start a new sentence if you want to use however since it is properly used at the beginning of a sentence. You can continue a sentence using but.
Still is used when an action is being continuously done or when a noun is continuing to be a certain adjective. This is usually used when you need to express an opinion that might be contrary to someone else's. For example, "Runners are still cute!" or "I still hate him."
Here's an example that has all 3:
I kicked him. However, he still loves me nevertheless.