Fair enough.
cp means Control Point. This means either Attack/Defend or Push maps. On an Attack/Defend map, the blue team is attacking and red is on defense. Blue is trying to capture control points while red is attempting to defend them. You capture a control point by standing on it for a certain amount of time. More people on it, the faster the capture happens. Red can block captures either by killing everybody who attempts to capture or by also standing on the point. Blue team wins by capturing all the points before time runs out. Red team wins by defending until time runs out. On a Push map, there are five control points. 2 on each side are controlled by the red or blue team, with the point in the middle neutral. The goal is for a team to capture all five points.
ctf means Capture the Flag. Both sides have a briefcase of intelligence. Go capture it and bring it back to your base.
pl means Payload. The blue team must push a cart along a predetermined path to the end. The red team must stop them from pushing. The blue team pushes the cart by standing near it. The more blue team standing near it, the faster the cart gets pushed. Red blocks them from pushing by killing everybody pushing the cart or by standing next to it. Blue wins if the cart reaches the end of the track before time runs out. Red wins if they prevent the cart from making it to the end before time runs out.
plr means Payload Race. Both the blue and red teams are attempting to push a cart to the other side of the map. Pushing a cart works exactly like it does in Payload.
koth means King of the Hill. There's a neutral capture point in the center that is initially locked. Capturing it works exactly as it does in Control Point. Both teams have a timer set at 3 minutes. When a team captures it, their timer starts counting down. The team that has the point is now going to try to defend the point while the team that doesn't is attacking. If the other team captures it, their timer starts counting down. The other timer is frozen in place (ie, if the blue team captures the point in the middle and holds it for 1:30 before the red team captures it, when they next capture the control point, the timer will start counting down from 1:30). Other than Arena, this is the game mode that is most like a straight-up deathmatch.
I don't remember Arena's acronym, but whatever it is, it means Arena. Two teams that the computer forces evenness on. You have one life. Kill people. Don't get killed. Your team wins either by eliminating the other team or by capturing a single control point.
Added at: 08:06
There are several common team sizes: 6, 8, 9, 12, and 16. The less people there are, the more important it is for you to kill and not get killed. Most pub servers have 12- or 16-man teams.