"Making a commitment" generally means promising to do something. For example, if you promised to go to a party or event, you can say this:

To be honest, I don't really want to go, but I made a commitment.

For a couple who are dating each other, "making a commitment" usually means deciding to get married.

I don't think I'm ready to make a commitment.

People especially use this phrase when talking about a man is ready to ask his girlfriend to marry him.