An ability is a term that covers all the methods that a character can use to influence the outcome of combat. These include actions (attacks, spells, etc.), morale (morale abilities), and tactics.
All three ability types (actions, morale and tactics) are gained in two fashions (except tactics which are gained in 3 ways).
One set is gained based on your career as you level up, meaning that a basic package of attacks, spells, tactics etc. is available to each career according to level. You can, conceivably, fill out your bars, tactics and morale slots using just these if so desired.
Another set of abilities can be acquired through various means such as masteries, based upon your race, based upon completion of various PvE or RvR goals, etc. Because some of these are gained exclusively of others, most notably as mastery options, it is impossible for a single character to possess all the abilities that are technically available to his career.