We sometimes get a little carried away with all the foreign words, not realizing that we are talking to a new player. If that happens consult here, or just ask us to speak normally :)
Aji - a potential. some weakness to be exploited
Atari - danger of immediate capture (stone or a group has only one liberty)
Dame - a neutral point not surrounded by either player
DDK (double digit kyu) - a player below 9k
Fuseki (The opening) - first few moves of a game (until a fight breaks out or players move above 4th line)
Geta - a net
Gote - the opposite of Sente, playing a move that does not need to be answered
Joseki - a known sequence (usually near a corner) that is believed to be locally optimal for both players
Miai - two or more different moves to achieve the same desired result. This usually means that you can safely play elsewhere.
SDK (single digit kyu) - a player stronger than 10k but weaker than dan.
Sente (initiative) - a state where you do not feel the need to respond locally and chose where to play next
Tesuji (clever play) - usually a beautiful local move that solves some trouble, or inflicts it, without any chance of being foiled
TPK (twenty plus kyu) - a completely inexperienced player
For more look here: Common terms
For even more look here: All terms