Humanitarian OpenStreetMap (HOT) activates mapping projects with on-the-ground partner organizations as both in response to disasters and as preparation for potential disasters. Links to those projects and instructions on how to assist in those efforts are available on the HOT Tasking Manager.
Information about specific mapping events and mapathons are available on the OpenStreetMap Event Calendar.
Experienced OpenStreetMap contributors can find mapping tasks in need of verification on Map Roulette.
For new mappers, LearnOSM has a guide specifically for getting started using the HOT Tasking manager.