A movement is a short section of a longer piece. A symphony, for example, usually has four short sections. The Fireworks suite has six movements.
A suite is a collection of something. If you go on a trip and stay in the Embassy Suites, you don't just get one hotel room, you get a suite of rooms -- a group of rooms. In music, a suite is a group of shorter pieces - you will often hear a suite of dances of a particular type. If you go into the Wang center or somewhere else to watch Tchaikovsky's Nutcracker, you are not going to see the Nutcracker Suite, you are going to see the Nutcracker -- a ballet. If you go to a Boston Symphony concert and they play a group of pieces from the ballet, then you are listening to the Nutcracker Suite - a musical piece that Tchaikovsky put together from some of the music from the ballet, intended to be listened to at a concert without the dancers.