Most heroes have a resource that limits how frequently they can use abilities (in addition to individual cooldowns). A hero cannot use an ability without having the required amount of his or her resource.
Mana[]
Most heroes use mana. It has a maximum of 500 at level 1, plus 10 per level up, to a maximum of 790 at level 30. It regenerates constantly at a rate of 3 per second at level 1, plus 0.0976 per level up, to a maximum of 5.8304 per second at level 30.
Heroes can pick up regeneration globes that restore 8% of maximum Mana over 5 seconds.
Brew[]
Chen uses brew. It has a maximum of 100, and regenerates only while using his trait, at 40 per second.
Fury[]
Sonya uses fury. It has a maximum of 100, and doesn't regenerate. Instead, Sonya gains 6 fury from each basic attack, and 2 fury each time she takes damage.
No resource[]
Abathur, Illidan, Murky, and The Lost Vikings have no resource. Their abilities are limited only by cooldown or other circumstances (e.g. Abathur only has damaging abilities while using Symbiote).
Patch changes[]
(Patch July 12, 2016) Note: Regeneration Globes: Previously: Restore 32 Mana over 4 seconds.; Now: Restore 8% of maximum Mana over 5 seconds.