Please explain the difference between a 'leveling compressor' and a 'compressor'

A compressor can be used for things like a snare drum, where you need finite control over the attack, threshold, release, and ratio. There are moments in your music when you need full dynamics (i.e. no compression), but other times when you wish to "kick" in the compressor at a transient when you want to thicken it up, but not ruin the impact. For example, a very dynamic vocalist might need some heavy, but fast compression because he is erratic on a mic, or a kick drum that varies wildly in impact. Another example would be when you want to compress only the loudest signals, but leave quieter signals below the threshold uncompressed. 

A leveling amplifier (or leveling compressor) is employed mainly when you want to add a compressed "sheen" to the music, best used judiciously. It's typically a compressor that has a low threshold and high ratio, medium attack, and maybe a slow release. Mix magazine states "it's a compressor that's always active" no matter how loud or quiet the source audio and "good for many sources". Normally, you might use one on a great vocalist, or a whole mix to bring up the level, but not incur a serious pumping of the audio. 

I deem them both really the same, but a leveling amplifier can sometimes be used to give your music a more broadcast radio appeal with fewer controls. The "UAD 2A" type is a perfect example of this type of compressor. It has two main knobs. Everything else it does for you on its own.

What I recommend is to use a standard compressor on specific tracks, and try a leveling amplifier (compressor in the plug-in world) for finalizing/mastering your mix. Again, use judiciously. Have fun!