Jayce, Nidalee, and Udyr are all three good examples of this. Their ultimates are all on a really short cooldown but they gain access to brand new abilities. League of legends guides can be some of the most extensive and informative guides out of any game out there!
There are three types of damage in the game, physical, magical, and true damage. How much of these depends on the champion that you are playing, if you are playing a champion that does physical damage then it is called an AD champion and a champion that does magic damage is called an AP champion.
Some champions have abilities that lets them bypass through all types of resistances and these champions deal true damage. True damage ignores any resistance or ability that the enemy has and this is the purest and highest form of damage possible.