Three pages of mostly good stuff but I still feel a need to chip in.
Profpointy covered the different types of cycling shoe in his
post.
The three-bolt cleat system ("SPD-SL" - of which your shoes belong to) are usually used by road cyclists that are into competitive cycling. Most cyclists I know, including some hard-nosed road cyclists who put in some serious miles, use the two bolt "SPD" style cleat. The SPD style generally are much easier to walk in, the cleats don't wear out so quickly (especially when walking), tend to have more "float" (the amount of lateral movement without the cleat disengaging - this can make a big difference to comfort) and the range and styles of shoes are much more varied; some look like trainers, I have a pair like walking boots complete with a walking sole. I've never used three-bolt systems but I get the impression that the two-bolt systems tend to be easier to engage and disengage.
Some people use cleated shoes because they want maximum transfer of power throughout their pedal stroke, using stiff shoes and little play - there is some contention over how much benefit cleated shoes actually provide in power transfer but they're the norm in competitive circles.
Some people use cleated shoes (me included) because they like the security and feel of shoes that are not going to lift or slip off the pedal. I'm not bothered about maximum power transfer - I like to spin the pedals with what seems like little effort.
I would advise anyone getting into cycling and wanting to try a cleated shoe, to go with the 2 bolt SPD style (except for serious racing cyclists - but they wouldn't be wanting my advice). It's a shame that you've already bought the 3 bolt shoes - you could use an adapter but as mentioned up-thread, they add considerably to the depth of the sole and are still more problematic for walking in.
Terminology can get confusing: we call the sort of cleated shoe that you clip into the pedal a "clipless" system. This is because in the old days, people used to use toeclips with straps to secure their feet to the pedals - these new-fangled things did away with the clips and so were called clipless. At the time, mountain biking was gaining popularity and riders were not finding it easy to quickly release their shoes from the pedals with clips and straps (if you cinched the strap up, you had to get your hand down to the pedal to knock the buckle to loosen the strap). The 2 bolt Shimano SPD was first on the market and was aimed at the mountain bike crowd. They introduced the 3 bolt system for the road racers. Although many cyclists of various disciplines now use the 2 bolt SPD, the system is still often referred to as a MTB system and the 3 bolt SPD-SL as a road system. This is particularly the case with shoes - many "MTB" shoes are not really MTB specific - they're just 2 bolt SPD specific.
Another confusing part of terminology is the pedal. Some SPD pedals have two clipping mechanisms - one on each side of the pedal, so it doesn't matter which way up the pedal is when you put your foot on it. Such pedals are "double sided". But you can get single sided SPD pedals with a flat or platform on the reverse side for use with non-cleated shoes (even crocs). Some people have come to refer to these pedals as "double sided" because they have two (differently) functioning sides.
As time goes on, more and more people use clipping or clipped to describe clipless pedal systems, and more and more people refer to single-sided SPD pedals as double sided. I even do it myself sometimes (if I didn't, some conversations would get really confusing).