You have two issues: the architectural style, and the ownership model.
A "condo" is never an architectural style. "Condominium" (to spell out the full word) is the legal term for units in a multi-unit building that are wholly owned (in fee simple) by their occupants, who all also have a partial ownership interest in the common areas of the building itself. Condominium developments are run by volunteer resident boards; the board members are elected by the residents, and all of the residents pay "fees" for the maintainance of the common areas and also usually the salaries of the folks who work in the office if there is one.
Cooperative apartments (aka "co-ops") are a different ownership model; these are most common in NYC. In this one, the building is owned by a corporation, and all of the residents buy shares in that corporation. Technically what you then own is a percentage interest in the building, rather than so many square feet of the inside of it.
A rental unit is just that: you pay for the privilege of occupying the space, but you have no ownership interest of any kind in it. Maintenance is covered as part of your rent, and you don't have to (or get to) make any decisions about how it is being done.
The terms for the various architectural styles can vary all over the place, but the terms for what kind of financial interest you have in the place are standard no matter where you are in the US.
You'll often hear people mention "renting a condo", but normally that doesn't mean a long-term rental (unless the developer who built the building is having real trouble selling the units and decides to rent them as apartments to help cover their debt.) Developments that are condos are marketed as more stable than rental developments because the occupants have an ownership interest in the common areas and will not be transient. Of course, in resort areas you have the phenomenon of condo buildings that allow the unit owners to rent them out short-term. This is done because those buildings are not meant as primary residences, and it is easier to talk someone into buying a second home if you will allow them to rent it out when they are not using it, so as to help cover the mortgage cost.
One other note: Traditionally, the nicer "senior communities" have required residents to buy their units, and the mortgage is held by the company that developed the community. Prices were very high because you couldn't shop it around, and the mortgage cost also included an automatic buyback of the unit when the occupants died or entered a nursing home, plus the monthly maintenance fees for the common amenities. Since the economy tanked and the housing bubble burst, that ownership model has really suffered, and a LOT of over-55 communities that used to require a very expensive upfront buy-in will now lease units at a monthly rent. Stinks if you bought in a decade ago, but times change.
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