Currently, there are three types of spetsnaz units known: Spetsnaz FSB (counter-terrorist and anti-deversant), Spetsnaz MVD and Spetsnaz GRU (army spetsnaz).
Spetsnaz FSB has two groups of special forces, "Alfa" team (A) and "Vega" team (B) (formerly used to be known as "Vympel"). The Alfa team specializes in counter-terrorism, and it is one of the most respected CT units in the world. The Vega team is a relatively new squad, made of members of the original "Vega" squad (that used to belong to MVD, before the USSR demise), and veterans of "Vympel", one of the most feared saboteur group in the Cold War. The new Vega squad is now responsible for anti-diversant operations, mainly concerned with the security of atomic energy plants.
Spetsnaz MVD has some Spetsnaz of the Interior Troops units, like "Vitjaz", "Rus", "Rosich", "Skif", "Grom", etc. Basically you can think of them as Russian Rangers. They are highly trained, well equipped troops of army spetsnaz soldiers. For example, the unit "Rus" had participated in many combats in Chechnya, and its few casualties were justified with hundreds of Chechen rebels killed (approximately, 1 to 200). Their missions can vary from just reconnaissance missions, to an assault, or even they (especially "Vitjaz") sometimes serve as the back up team during the counter-terrorist operations of Alfa.
Spetsnaz GRU are considered the most secret units and most skilled soldiers of the Russian Army. They are shadow warriors of GRU (National Reconissance Agency). Not only they are one of the best infantry in the world (in the mid 1990s, there was a competition in Alaska, USA, of Spetsnaz GRU and US Rangers. Rangers managed to be first only in one competition - lifting weights. Spetsnaz GRU was better in any other aspect, including running, target shooting, stealth, etc), they also have one of the best combat divers in the world. The units of spetsnaz GRU have no official names, such as of Spetsnaz MVD. So, they have only numbers, for example, "18th Brigade of Spetsnaz". Little to nothing is actually known about the operations of Spetsnaz GRU, but it is known that the units were heavily participating in wars in Afghanistan and Chechnya.
Rumors say that the most secret agency, SVR (Sluzhba Vneshnej Razvedki, Russian analogue of American CIA) has also its own Spetsnaz unit, called "Zaslon". but nothing is known about them so far, and of course the Russian government will not acknowledge the existence of this group officially.