It's a lazy way of saying 'give me'. We'd usually only use it in spoken English, and when you're talking to someone you know well, as it's very informal eg 'gimme a piece of your chocolate'.

Just like Katharine has just explained you, people are totally lazy sometimes and they like to make the words much shorter. For example: 1. Gimme - Give me (for example: "gimme a kiss" instead of "give me a kiss" ) 2. Gotta go - I gotta go (For example: "I have to go home" instead of "I have to go home" ) 3. CYA - See you (For example: " Cya tomorrow " instead of " See you tomorrow) I hope I made it claer now to you. If you have any futher questions, feel free to ask me.