Freehold
Full, perpetual ownership of the unit and a share of the land, registered in your name on a DLD title deed. Available to all nationalities, but only inside designated freehold zones (e.g. Dubai Marina, Downtown, Palm Jumeirah, Business Bay, JVC, Dubai Hills). This is what most foreign investors are buying.
Leasehold
A long-term lease — usually up to 99 years — granting use of the property while the freeholder retains the underlying land. Common in some older or master-developer areas. At expiry the property reverts to the freeholder unless renewed; the remaining lease term materially affects value and mortgageability.
Commonhold
The structure that sits under most apartment buildings: you freehold-own your individual unit and jointly own the common areas through an owners' association, which levies the service charge. Your title is freehold; the building is governed collectively.
Usufruct / Musataha
Registered long-term rights (usufruct = right to use, up to 99 years; musataha = right to build/develop, up to 50 years, renewable). You see these on some commercial and master-planned plots rather than typical residential resale.