A tempo marking refers to how fast the piece is to be played. Sometimes the accompanist (the person playing the piano) will ask you what your tempo should be, feel free to go over and work it out with them.
Beats and Measures
A beat is a single unit of music. We measure these in groups of two or three. Those groups are what we call "measures" or "bars". When an audition calls for 32 bars of music, they're asking you to count those measures.
The melody is the main musical idea, in your average song the melody is the part with lyrics.
Your starting pitch is the first note you sing in the song, when you are singing with live accompaniment you can usually ask for your starting pitch so you start singing in the right key.
Intro / Vamp
If an accompanist asks if you want the intro, they're asking if they should play a few measures before your audition cut, so you can come in confidently. Sometimes this means playing a vamp, which means the pianist will repeat one or two measure of music until you start singing.
A musical key determine the range of your piece, so be sure to double check your sheet music and make sure you've got the key you need, otherwise you won't be able to sing along when the pianist plays it.