Goal Attack is a multiplayer game mode. You compete to complete the selected goals from Classic Mode. This mode was first introduced in Tony Hawk's Pro Skater 4, which in THPS4 you have to unlock the challenges to be able to play them. In the later games such as Tony Hawk's Underground 2 and Tony Hawk's American Wasteland since they already have Classic Mode you can use those goals. You can also use Create-A-Goal to use custom goals made by players. In Tony Hawk's Underground if you can use vehicles for custom goals, if you use a custom goal with vehicles online in Goal Attack, with collision enabled you can fling you and your friends and it is hilarious.
In Tony Hawk's Underground 2's version of Goal Attack you get to use the goals from Classic Mode, again you can create your own goals with Create-A-Goal, there are no vehicles this time though. You get to use Classic Mode goals in Tony Hawk's American Wasteland's Goal Attack as well but because no maps from Story Mode transfer over to Classic Mode the developers of the game had to substitute by making unique goals for the specific maps.