the three major coral distinctions you will see are generally "SPS" "LPS" and "soft"
though Small Polyped Stony and Large Polyped Stony are poor desingations because some of the SPS have bigger polyps than LPS and vice versa; this distinction is actually based more on light and feeding requirements than anything else. though truest classification is "stony" and "soft"
generally speaking stony corals of the so called SPS variety need high water movement, very intense lighting, and very clean water. Many of the LPS corals, however, thrive with less lighting and water movement and will do ok with a little bit of disolved organics in the water. softies (leathers, mushrooms, xenia, zoanthids, etc) will generally use the least amount of light, need the least water flow, and tend to be the most hardy. of course, there are exceptions to every rule.
generally, if you buy an SPS coral frag it will come attached to a piece of LR or putty plug. then you just stick that in your tank in the appropriate place. LPS corals tend to have large skeletal structures (look into Euphyllia, Trachyphyllia, Lobophyllia, etc) which you can set into your rockwork and will not come attached to anything. softies like mushrooms, zoanthids, GSP, and xenia (often called colonial because they grow on a big mat) will generally come attached to a piece of live rock rubble and will then spread to cover that rock plus the surrounding rock. leathers and corals like nepthea generally come attached to a piece of live rock rubble but will not spread onto other rock, they just get bigger. you can "frag" leathers and such and cut them into pieces and grow new ones. a lot of people do that.
hope that helps some