It'll vary, of course. Often condos will have both a numeric limit (for instance, 1 or 2 pets) and some other restriction, such as size (for instance, weighing no more than 25 pounds). In cases like that, both cats and smaller dogs would be permitted. If pets are permitted, there often are some additional limits. Sometimes (applicable to dogs) certain breeds aren't permitted, such as pit bulls. (I'm not debating the merits of such rules, just noting that they exist.) And the rules often will provide that if the pets attack other pets or humans, the condo board has the right to request their removal.
What you have to do is read the condo documents very carefully to see what is and what isn't permitted.
Hope that helps.