To "commit to" something means that you make a strong promise (to yourself or to other people) that you're going to do something. To express who the promise was made to, use the phrase "make a commitment to (someone) that (clause):
I made a commitment to myself that I'd own my own business by the time I turned 30.
We've made a commitment to our customers that we will always deliver the best quality at the lowest prices.
A commitment that you make to yourself is basically just a strong decision that you make. A commitment that you make to other people is a promise or, in some cases, a legal contract. Some things that people often "commit to" include:
commit to a relationship (with your boyfriend or girlfriend)
commit to a project
commit to a schedule
Notice that "commit to ___" is different from "commit ___", which is often used to talk about doing crimes or other bad actions:
commit adultery (cheating on your husband or wife)