When you request something, or when you make a request, you're asking for something that you want, and you're asking in a nice, formal, polite way. Imagine you're at a fair, and a worker asks you, "What kind of balloon animal would you like?" You might request a purple cheetah by saying, "May I please have a purple cheetah?"
"Request" is a serious word. We request things from workers, from companies, from teachers, from doctors, from librarians, and so on. But most of the time, we don't "request" things from our friends and family: we "ask" instead!
At a restaurant, you could request extra tomatoes on your salad. You could request extra napkins, or a high chair for your baby sibling. You might request some cherries with your drink; if your server says "Sure!" and brings you the cherries, then your request has been granted.