For the U.S. at least:

With condos, there's a condo association that owns all the common areas. Then the association itself is owned by the owners of the units, and the management is elected by the owners.

With co-ops, the unit owners directly own the common areas in common, and the management is also elected by the owners.

Functionally speaking they're very similar, and co-ops tend to exist in places where this legal structure predates the invention of homeowner associations (basically New York).