"Condominium" is a term popularly used to describe an apartment-style home that is owned, not rented. More accurately it defines a specific type of ownership of real estate, where any style of home is owned "from the studs in" and the ownership of the land is shared by all homeowners who are part of the condominium association.
Owning a home "in fee" or "in fee simple" means you have ownership (or in some cases, shared ownership) of land, as well as the actual home.
In contrast to owning a home "in fee," where you have ownership of (or shared ownership of) the land, someone else retains ownership of the land when there is a land lease situation. The homeowner pays rent to the land owner under the terms of the land lease, and ownership of the land is never transferred to the homeowner.
A townhome - also sometimes called a row home or a townhouse - is a home that shares a side wall with at least one other home. It may be have one, two or more levels, but would not have another home above or below it. Townhomes are most often (but not always) condominiums.
Depending on local custom or community definitions, there is no universal definition for this home type. In The Peninsula, a villa is a detached, single-family home that is owned as part of a condominium association. The homeowner owns the home "from the studs in" and has shared ownership in the land.