The alternative advancement system is a non-level-based advancement options for those level 51 and above. It was originally one of the features of EverQuest: Shadows of Luclin.
There are a very large number of alternate advancement abilities, which you can buy with AA points. In order to attain these points, you can toggle your experience into the alternative advancement pool. (This can be done by hitting the default of V, or by going into the inventory window and using the 'Alt. Adv' button.) When you do this, your experience will be allocated to this pool in lieu of your "leveling" experience bar. Switching between the two pools will not cause you to lose experience and experience loss due to death suffered by the character will always be taken from the "Leveling" experience pool, and will never impact your alternative advancement pool.
The AA window is reached through the inventory window. Every class has their AA's separated in to three tabs, the general tab that all classes share, the archetype tab which each archetype (fighter, caster, priest, hybrid) shares, and the class AA tab, which is specific to each class and described on their respective pages.
For a general overview on various strategies for developing your character's Alternate Advancement abilities, take a look at this guide on Alternate Advancement leveling.
For reference, the non-editable spreadsheets used to consolidate the information on the wiki pages is available at http://www.editgrid.com/explore/user/yther_ore/AA3_Ref for ranks 1-36 and http://www.editgrid.com/explore/user/yther_ore/AA3_Ref_Ext for ranks 37-72.