these are known as 'mums' in the hort. trade or just casual gardeners as well. There are two kinds (to keep things simple). One is the annual/
greenhouse type which you buy for Easter/mother's day. The other is the perennial kind which can be put in the ground. To plant in the ground, you should loosen the bottom roots. Ripping a few out will not hurt the plant. You should also score the root ball vertically with a knife about and inch deep before planting. You should definitely repot after this flush of
flowers fade. If you are diligent in keeping on top of your deadheading (snipping off dead blooms) which I never am XD, and trim appropiately, your mums might give you another flush of blooms if they're the annual kind. I'm not too sure about the perennial kind though.