There are several uses, but by far the most common is to show possession. Think of it as an apostrophe s. So, if 本 means "book", and マリ is your name Mari, then you can say "Mari's book" like this: マリの本

ありがとございます~

In case you don't know the characters - Yuta is showing "Watashi no namae wa Mari desu" Watashi no namae (My name...) The no is showing that the name is yours. "Watashi wa America-nin desu" I am American. You wouldn't have possession ("no") here.

My name is Mari. 私の名前はマリです。 I'm American. 私はアメリカ人です。