African violet or Saintpaulia Hybrids 
Family - Gesneriaceae

There are many variaties of the Saintpaulia Hybrids with different colors: pink, white, purple, blue. You can also have multicolored flowers (as the one I have on the picture here), double or single, with ruffled edges etc. Some people think that it is really hard to keep the african violets healthy, flowering and nice looking. From my experience - not at all. There are couple of important things that need to be taken into consideration:
- don't put violets in the sunny location, under direct sun, especially in the spring or summer - the plant will burn in no time, you will be left with brown patches all over the leaves. I have really good results with north facing window - it is bright enough but never sunny.
- normally african violets flower all year long with short periods of rest. Now, if you will move the plant to a larger pot - you would have to wait for a while before it'll bloom again. Plant will try to fill all the pot space with roots first and only after that - you will get the flowers. So, normally when repotting, I keep the same-size pot, just use the fresh soil. Unless you want bigger plant, I mean the size of the bush. If this is the case, then definitely take the larger pot.
- and the most important point - never ever water african violets from the top. Provide it with deep saucer, pour water into the saucer, let the plant drink for a bit and then remove the extra water from saucer. And one more thing - use the water of room temperature, cold water will damage the plant.

As the propagating concerned - it is the easiest part. Cut the leaf from the plant , put it into glass with water and wait. In a week or so you will see little roots starting from the leaf, when they will reach significant size, plant it into a pot . For african violets you can use all-purpose soil; if you want to go fancy, you can buy soil mix labeled specifically for "African violets". I tried both, didn't see much of a difference.