Uramaki is considered western-style sushi. It's basically maki that's inside out. The rice is on the outside and the seaweed is on the inside. Like maki, uramaki can consist of pretty much anything. Many "special" rolls will be made uramaki-style.
Maki is rolled sushi, and probably what you think of when someone mentions the delectable Japanese dish. Fish, veggies, or other ingredients are rolled in rice and wrapped with seaweed. A roll will consist of six pieces.
Sashimi consists of thin slices of raw fish such as salmon or tuna. It is not served with rice, but is usually be served on a bed of daikon radish. When you order sashimi, expect a plate of various types of sliced raw fish.
Nigiri consists of thinly sliced, raw fish pressed on top of sushi rice. There are some forms of nigiri that have cooked or seared fish, if you can't bring yourself to eat it raw.