I have a few things to do today, so I'll go home early.
母に頼まれた用事で郵便局へ行った。
I went to the post office on an errand my mother asked me to do.
I have a little something to take care of with him, so I'll excuse myself first.
I'm going to the bank to take care of something.
I got my errands done in the morning.
Something urgent came up, so I couldn't go to the party.
My mother asked me to run an errand, so I went shopping.
用事 refers broadly to a task or business one has to attend to, while 使い specifically means going somewhere to run an errand for someone else.
用事 is a casual word for a personal errand or task one has to do, while 要件 sounds more formal and is used for business matters requiring attention.