This idiom, while being a small one, is very powerful. It means that the decision belongs to the person something is "up to." If someone says a decision is up to you, then it is your duty and responsibility to decide in the particular situation. If you don't want to decide, you have to refuse or reject the offer if possible.
Example 1: The decision about when to move on to chapter 2 will be up to you, so tell me when you are ready.
Example 2: John's boss said that it would be up to him whether John continued working at the company or not.
Example 3: Do you want Italian or Chinese food tonight? I'll leave it up to you, because it doesn't matter to me.
Example 4: After my son becomes 18 years old, the major decisions of his life are no longer mine - they are up to him.