VDV (Vozdushno-Diesantnye Voiska) – while they might be described simply as „Airborne”, this does not represent the pride and eliteness of these troops. “Diesant” can be described as “landing troops”, those who jump in into a middle of action – there are naval diesant (marine and the like), the infantry deployed from armored carriers is referred to as diesant (like their predecessors in the WW II, riding piggy-back on Soviet tanks), and of course there’s airborne diesant. So, the translation closer to the soul would be “shocktroops”. VDV are chosen, trained and equipped to serve to these expectations – they train hard, party hard, and fight hard. They also formed the core of many Soviet military intervention forces – Airborne were to spearhead the attack.
VDV were – in USSR times – an army within the army. A separate combat arm almost. Up to the point where they had heavy equipment being designed and produced exclusively for them – e.g. BMD series of IFV – pararel to BMP series IFV, but lighter (for air portability and paradrop deployment). Nowadays, they’re being rebuilt back to (and past) that strength.
If looking for an analogy, try imagining US Marine Corps’ Russian edition – except that these guys come from the sky, not from the sea.

Alfa (Alpha) – Departament A of the Special Operations Centre of FSB (once called KGB, the Russian internal security and intelligence) is a counter-terrorist unit, similar in profile to US Delta Force. They specialize in hostage rescue, but can undertake most SpecOps tasks, and are equipped to the top world standards.
Alfa has about 700 operatives, divided into 5 operational groups.

Vympel (Pennant) – department V of SOC, trained for operating in industrial complexes of special importance (they’re the guys who will be used to storm a nuclear power plant taken by terrorists, or something the like). However, they have broad general training, and are often used hand to hand with Alfa. This unit is smaller, with about 500 men and 4 operational groups.

Note on FSB: the Russian “(anti)terrorist law” entitles FSB to use their troops (as well as Spetsnaz and Osnaz forces leaned from elsewhere for that purpose) to conduct counter-terrorist operations anywhere in the world, on the order of President.

Spetsnaz / Osnaz – the Russian SpecForces most people have heard about, but still there’s little known about them. Spetsnaz (Spechyalnovo Naznachenya, literally “Special Purpose (forces)”) are the commando arm of the GRU, Russian military intelligence. They have fierce reputation due to their harsh training methods, and therefore are considered to be extremely tough. They’ve been deployed for reconnaissance, sabotage and COIN operations.
Spetsnaz is usually placed within the structure of VDV (naval Spetsnaz – within naval infantry, i.e. marines), but not under their command – they wear the same uniforms and insignia, and are hard to tell apart (they are also likely to wear uniforms and insignia of any nearby unit, just to blend in and remain unidentified as Spetsnaz). They shall be considered an rough equivalent to US Green Berets or Army Rangers. Naval Spetsnaz is trained as combat swimmers /frogmen, and do utilize minisubs. Each Fleet of the Russian Navy has a Naval Spetsnaz brigade assigned to it. The number of Spetsnaz units within VDV structure is unknown, however, it is estimated that there might be as much as two dozen independent Spetsnaz brigades.
Osnaz (“Separate Purpose”) is their equivalent, subordinate to the Ministry of Internal Affairs.

Note on spetsnaz: the term is often used as a catch-all phrase, literally an equivalent to English term “special forces” – it can mean anything from an elite paratroop unit (or even a police SWAT-type team) to the true black ops detachment. All that adds up to the Spetsnaz’s mystery.

Gvardieytsi (the Guards) – the title of Guards have been applied to many units in the history of Russia and USSR – a few have been granted this at their creation, most however were awarded for extraordinary bravery in combat. This high status – a mark of the elite among the Russian army – was later kept up with the best equipment available (or just being issued to these troops first), best recruits that were there, and high standards of training and discipline. There is a reputation behind this title, and soldiers do a great deal to act up to it.
Whereas there are several armored, mechanized or air Guards units, and that means merely that you are dealing with an elite panzer, infantry or Air Force unit, there is also a number of Airborne Guards and special purpose Guards units. They do carry traditions of the units that’ve distinguished themselves in combat in the wars of the past, but this often also means that you’re dealing with a true Spetsnaz unit. A tiger in wolf’s skin.

Razviedniki (the Recon) – there are many reconnaissance forces in the Russian army. Most are just that: advance troops. However, there are also “long range reconnaissance” companies (on a division level), or even battalions (army-level). This can mean anything from poor man’s spetsnaz, to true Military Intelligence’s Spetsnaz attached to the unit. Moreover, “razviediniki” is one of the army’s terms to describe “the guys whose true name is unknown and better do not speak about them too much” (other known terms are “raidovniki” – raiders, and “okhotniki”- (game) hunters).