The following common terms all apply to Association Croquet.
Lift. If a player runs hoop 1-back or 4-back in a turn, his opponent is entitled to a lift. This means he can lift either of his balls and play them from any position on one of the baulk lines if he so chooses.
Contact. If a player runs hoops 1-back and 4-back in the same turn, and his partner ball had not run 1-back at the start of the turn, his opponent is entitled to a contact. This means he may either take a lift (see above) or lift his ball to any of the other 3 balls and immediately take croquet from that position. As a contact concedes control of the game to the other player, conceding a contact is generally avoided at all costs, which means that a player is likely to stop on his first break before running 4-back.
TP or Triple Peel. One of the hallmarks of a good player, the triple peel allows a player to complete the game in two turns without conceding a contact. This is done by peeling the partner ball through its last 3 hoops and pegging out to win the game. TP is often seen in score lines - e.g. Fulford beat Clarke +17TP would mean that Fulford won with a margin of 17 hoop/peg points, and completed a triple peel to do so.
TPO and OTP. A TPO is a Triple Peel on Opponent, completed on an opponents ball. In other words, the player has peeled his opponents ball through its last 3 hoops and pegged it out in order to remove it from the game. If a score is reported as a TPO, this means the player who completed the TP then went on to win the game. If the score is reported as OTP this means the player who completed the TP went on to loose the game.
SxP or Sextuple Peel. An advance on the triple peel, this manoeuvre is only generally completed by the very best players. In order to avoid even giving his opponent a lift, the player stops his first break before running 1-back. This leaves a sextuple peel to complete, which means in his next turn he peels his partner ball through its 6 remaining hoops while taking his second ball to the peg, and pegs out to win the game.
A full list of technical jargon can be found on the site.