I know this has been answered but I will give my 2 cents.
Understudy - They are usually ensemble members who learn a principle role in the show. They may take over a principle role in the show if the person is on vacation or sick. There might be more than one understudy.
Swing - These people don't normally perform in the show. They may fill in for a vacationing/sick ensemble member and may even have to perform a principle role. They sometimes have to go on at a moment's notice.
Standby - They are like an understudy, except there is not a place for them in the ensemble...they might not fit in because of age. They have to be ready to go on stage if an actor/actress has to leave before the show is over.
Alternate - They perform at certain performances. For example, if the role is quite demanding, one person may perform 6 of the 8 performances a week. The alternate will take on the other two. For example, in Phantom of the Opera, Julie Hanson is the alternate Christine who performs on Monday nights and Wednesday matinees, while Sandra Joseph performs in the other 6 shows.
"We like to snark around here. Sometimes we actually talk about theater...but we try not to let that get in our way." - dramamama611