Oh already nice answers, but just let me try.
What Samin and mog1mog said are correct.
わたし and あたし are used by women. So you use わたし or あたし. あたし is more informal.
If men say わたし, it's mostly formal situations, or they want to behave in a formal way.
All of them means I/ Me / my and mine.
They are usually followed by a small letter, like は, が, に, and の etc
For example :
わたしは わたしの かれに プレゼントを あげました。I gave my boyfriend a present.
わたし(は) わたし(の) かれ(に) プレゼント(を) あげました。
( ) are those small letters.
かれ or かれし means one's boyfriend in Japanese.
私のカレは私にプレゼントをくれました。 My boyfriend gave me a present.
In the 2 sentences, 私に is me. かれに is him.
僕 is kind of more gentle or soft way to say 俺 used by men, so if you use them, it would sound a little strange in general.
Now maybe you try to make a sentence? ^^
Okay, thank you.