To entrust is to give someone a responsibility you assume she will fulfill. If you entrust someone with the task of getting you to school on time, make sure she’s punctual.
To entrust is to let someone take care of something for you because you believe she will protect it. It could be a duty or a thing — you might entrust a nursing home with the care of your parents or entrust an accountant with your finances. Entrust is a verb that needs an object, so you always entrust with or to something. If a friend entrusts you with a secret, she trusts you not to tell.