That is to say, how would one describe the year they were born in? Or the year they got married?
Was it based off of consulships? Or was their an event that served as a marker, like Christ's birth is to us in modern times?
Was it based off of consulships?
In general, yes. That was the normal way of referring to a year. Such dates are generally listed as having occurred in the consulship of A & B or X & Y.
In the late republic people also started to use ab urbe condita (since the founding of the city) which they placed in 753 BCE. But it never entirely replaced the consul references.
There's always room for discussion, but perhaps the section Calendars and timekeeping in our FAQ will answer your inquiry.