Blitz events can either be solo, or Alliance events, where your whole Alliance works together to achieve awards. The goals can vary, but can include collecting/crafting Gear pieces, or collecting Hero Fragments of a specific type.
For every milestone, you get rewards, which you can claim from the Event screen. If you don't claim these before an event is over, you can return to the Event menu for a short time to collect what you're missing.
Note that unclaimed Event rewards will disappear after some time.
Bounties is an Alliance-only mode where Alliances compete on a global leaderboard by taking down their own specific targets. Every time you finish a mission, there's a chance to spawn a Bounty, which your Alliance needs to work together to take down.
Some Heroes may have boosted damage and health during particular Bounty events, and some Bounty targets may require certain Heroes to fight.