'Roman' covers a lot of time and geography, but assuming you are most interested in the late Republic and early Principate (like most people), I think you might find this recent answer of mine helpful. Also, /u/Ipikrates gives a great answer here about classical cavalry in general, with some excellent sources, and I followed up with an answer that, in parts, is more Rome specific.

The short version is that Roman cavalry was originally modeled on the Greeks, and assimilated tactics and armaments from other more eastern opponents as they encountered them. Their cavalry was overwhelmingly auxiliaries from outside Italy, and biased in favor of light cavalry missile tactics. But they were also early adopters of the horned saddle that made shock tactics practical. Over time, and especially in the East, these grew in importance, with the adoption of Persian style heavy armored cavalry. They did not class heavy and light types separately. The basic tactical unit was the alae, or wing, which referred to their normal tactical deployment on the wings of the legio. Alae were nominally either 512 or 768 troopers, commanded by a Prefect and divided into turmae of 32 men, each commanded by a Decurian. A Legion typically had two or more alae attached, and deployed on its flanks in battle. It also had a small force (around 120 men) of organic cavalry drawn from the legion itself. They scouted, rode dispatch and performed various housekeeping and policing tasks. In addition to the missile and shock tactics I describe in those answers, alae performed all the typical modern cavalry tasks. They screened advances and covered retreats, protected flanks, performed reconnaissance, counter-reconnaissance and raiding, and acted as police and guards at remote outposts.

I hope that is helpful to you.