Simple Definition:

The barter system is a method of exchanging [giving something in return for something else] goods or services with other people without using money.

For example, if you have a bag of oranges and you want a book, you can find someone who has the book and wants oranges. You can then exchange your oranges for their book without using money.

Very Simple Definition:

Barter system is when people exchange things they have for things they want, without using money.

For example, if you have a toy car and you want a toy robot, you can find someone who has the toy robot and wants a toy car. You can then give them your toy car and get their toy robot in return, without using money.