What type of competitive gaming is available to the public TF2 Community?

The Official Team Fortress Wiki has names of some organizations that do league play. If you are in North America, the ESEA has an open division and pick up games for players of all skill levels.

These leagues may run the Standard competitive format, which is a 6 on 6 match with crits off and team composition limiting (only 2 scouts on a team at one time, etc.) and fixed weapon spread; Highlander, which is a 9 on 9 match with the same rules as SCF, but only 1 player per class per team; and fun competitions like Ultiduo or Basketball.

Due to the decrease in popularity in competitive play, the ESL shut down their TF2 ladders on May 5th, 2012.