Difference between 'part' and 'a part'?

When part is used without the article it has the meaning of section or segment - it is less than the whole object.

A part has the meaning of a specific sub unit. It normally isn't used for something which isn't an identifiable thing all by itself (though just to be confusing sometimes it is.)

Generally, you can use either.

'My arm is part of my body' and 'My arm is a part of my body' are both everyday uses to native English speakers. They mean more or less the same thing. There is a slightly nuanced difference in meaning but one which even a very experienced speaker would find it difficult to explain. So I wouldn't worry about that.

I presume in your first example it is meant to read 'This is a leg. It is part of your body'.

Do remember always to start a sentence with a capital letter. You will see that I have extensively edited the OP, but I didn't interfere with the sentence examples you gave.

Use "part" when it is not meaningful without the rest of the whole; use "a part" when it is meaningful and combines with other meaningful parts of the same whole. For instance: A leg is part of my body; but This item is a part of my collection.

If you say "Money is part of the American Dream", you mean that both ideas are strongly linked together. If you said "Money is a part of the American Dream", it would mean that money is only a part of the American Dream. You would focus on just one part as opposed to the rest. The second solution is grammatically correct but seems awkward as it would be dificult to find a context in which it fits.

I'm not a native too but i took this years ago in my English class We use part to describe something that don't have value or use alone without the whole combination or when the whole combination can still be with benefit without this part (as your leg) "A part" to describe something that have value alone and can be of benefit alone without the whole combination ,like a doctor in his staff, or when describing a part that the whole combination won't have any benefit without (as the neck) Some people live without a leg, but none lives without a neck