Perhaps one of the most exciting facets of the Russian language — and one of the many reasons why it is my favorite language — is the logical construction of words using prefixes and stems.
When learning a new language (or anything else!) I think we learn best by creating a web of knowledge. Rather than spending hundreds of hours torturing myself by memorizing vocabulary, I prefer to learn how the language words and use that to build a "system of crutches", so that when I forget a word, I can figure it out from what I know instead of being frustrated and giving up.
In Slavic languages like Russian, it is super-easy to quickly build vocabulary by simply learning stems and prefixes. It's even handy in a pinch, when you don't know a word for what you want to say, because you can just attach the appropriate prefix to a word stem that you already know, and you'll be able to communicate your thought effectively even if it's not the right word.
UPDATE: The best and most comprehensive online Russian course I've found is Rocket Russian (I've had great success using it). If you're learning Russian and want to get your head around prefixes, then I recommend it.
Unlike the rag-tag collection of prefixes in the English language, the Russian prefixes are complete and clear, and rather well-defined. I like to think of each prefix as a leg of a journey, and I've come up with a method of visualization that I believe makes learning Russian 1000-times easier.
In the image below (click for a larger version) you imagine you start from the center of the left circle and end at the center of the right circle. Everything that happens along the way can be described using one of the prefixes.
First, let's look at the beginning, and how we can describe going away from a place of origin.
от- : off, aside.
от- (aside) + ложить (to lay) = отложить - to set aside
вы- : out.
вы- (out) + ход (movement) = выход - exit
у- : away.
у- (away) + бить (beat) = убить - to kill
из- : from.
из- (from) + влечь (draw) = извлечь - extract
Next, we'll skip to the end and look at how to describe getting to a destination.
: approach; coming "up to".
под- (coming to) + сказать (to say) = подсказать - to suggest, to prompt.
: in. within.
в- (in) + ход (movement) = вход - entrance
при (arrival) + земля (the ground) + -ся (reflexive) = приземлиться - to land.
до- (reaching) + гнать (to chase) = догнать - to catch [up to]
In getting from the beginning to the end, there are a few things that can happen.
: across; over.
пере- (over) + садить (to put) = пересадка - transplant
про- (through) + водить (to carry) = провод - wire, conduit
на- (on) + лить (to pour) = налить - to spill
: drop in
за- (dropping in) + грузить (to load) = загрузить (to download)
воз- (up) + брать (bring) + -ся (reflexive) = взбираться - to climb
с- (down) + крыть (cover) = скрыть - to hide
: out in all directions.
раз- (into all directions) + лететь (to fly) = разлететь - to scatter
: in from all directions.
со- (from all directions) + брать (bring) = собирать - to gather
: around. about. (not pictured)
об- (around) + нимать (to press) = обнимать - to embrace
: by increment of. a bit of. (not pictured)
по- (a bit of) + есть (to eat) = поесть - to have a bit to eat