Jazz, perhaps more than any other genre of music except classical, uses a variety of dynamics. Not everything is played at the same volume. It's important to pay attention to the dynamics on the sheet music you are playing, as well as trying to match up with the rest of the players if you are playing in a jazz ensemble, unless a piece calls for you to be quieter or louder than the other players. Jazz is a genre of music full of nuances, and dynamics certainly contribute to that.
In addition to the notes and dynamics you'll find on sheet music, you'll find other clues and suggestions as to how a piece is played. It can be easy just to ignore these and begin playing the notes, but that can result in a failure to capture the spirit of a piece. Look for things such as "bouncing" or "nice and easy" to give you hints as to the style of play required. As well, pay attention to what tempo is listed. If you own the sheet music, consider writing clues on it for yourself to help your playing. Be sure to use pencil so you can easily erase if necessary.
Jazz typically swings, meaning the notes are often played slightly behind the count. For example, if you see two eighth notes, counted as "1 &," the second note will usually be played slightly later, so it'll sound like you're playing, if counting, "1 a." This gives jazz its characteristic "swing." Likewise, you might be asked to play on the beat, behind the beat, or ahead of the beat. These things mean just what they sound like -- slightly modifying the placement of the notes to obtain the desired feel and style the piece asks for.
A rhythm section usually consists of drums, bass, piano and sometimes guitar. A rhythm section mainly keeps time. It is a key component of any jazz ensemble. It's important that you listen to them so you can line up with their dynamics, tempo and feel. Essentially, they steer the ship, providing a framework in which improvisation (the essence of jazz) can occur. If you're in the rhythm section, be sure to listen very carefully to the other members so you can lock in with them; the best rhythm sections in jazz are often said to possess a "telepathic" rapport. The more solid a rhythm section is, the more solid the entire jazz ensemble will be.