I just installed TF2 and launched it. A steam pop-up said the app already was running.
That weird because I haven't played it on that computer. There is no task called team fortress 2, and the game doesn't seem to be running.
Try using the "verify integrity of game cache" feature in Steam. Should be able to find that if you right click your game, go to properties, and look under all of the tabs (if I remember right the last tab). If it still does not work try deleting the local content and re-downloading the game. P.S. Sometimes the simple solution is the best solution. Hopefully that works :D
All source-based games run off their "parent", Half-Life 2. As such, their process name in the Task Manager is
Open the Task Manager, which can be done by right-clicking the Taskbar or pressing ctrl+alt+delete. Both locations have options to open the Task Manager.
Locate the process
hl2.exe. It should be pretty easy to locate since it's going to be most likely the top resource-hogging process, or you can sort processes alphabetically. Once found, click on it and click End Task.
You should now be able to successfully launch Team Fortress 2. If not, it would be a good idea to reboot your computer.