Alone can mean ‘without anyone else present’ or ‘without help’.

I sat alone. (Nobody else was present.)
I went alone.
I did the job alone.
The house stood alone on the hillside.

Alone can follow a noun or a pronoun. In this case it means ‘and no other’.

Smith alone can solve the problem. (= Smith is the only person who can solve this problem.)

To leave / let somebody alone is not to touch or interfere with him / her.

Leave me alone.

Lonely

Lonely is an adjective. It means ‘without companions’. Lonely can also be used to refer to places without many people or which are not often visited.

A lonely village

When you are feeling lonely, you are feeling sad because you are alone.

I like to be alone for short periods, but after a few days I will start feeling lonely.