To be considered a 'voting' member in our organization, you have to be an active member, and also have attended a certain number of events in the past year. I see how to run a membership report, and I see how to run an event report, but how can I search for 'active member' AND 'attended 3 events in the last 12 months'?
There are several answers to this; I'll give you three! Use whichever works best for you.
Note that I originally thought that either Summary Fields or Event Count custom search extensions would be part of this solution, but neither can count only events in the last 12 months without modification.
Easy, best for now
Search for active members. Add them to a group. Run the event report, filtering by the Active Member group. If you need to run this report regularly, the Active Member group should be a smart group.
There's an alpha version of the Search overhaul that ships (disabled and hidden) in CiviCRM 5.28 (which released three days ago). This will allow doing searches like you describe in a single search pretty easily.