An ability is anything a card lets a permanent, player, or spell do beyond its raw stats. There are three main types: activated abilities (written as "cost: effect," like tap to add mana), triggered abilities (written with "when," "whenever," or "at," like "when this dies, draw a card"), and static abilities (continuous effects that just exist, like "creatures you control get +1/+1").
The big nuance people miss: abilities exist independently of their source once they're on the stack. If you activate or trigger an ability, then the source leaves play, the ability still resolves. Also, abilities aren't spells—so "counter target spell" can't stop them; you need something that specifically counters abilities.
In Commander, abilities are where the format lives—engines, combos, and value loops are almost all ability-driven. Knowing what can respond to (or can't stop) an ability decides countless interactions.