First off, the fact that everything you've ever heard on the radio has been through countless stages of compression should demonstrate just how important it is, at least on the recording end of things. There's also the fact that the radio station compresses the living poo out of the songs before they broadcast it. People who make a lot of money "stoking the star-making machinery behind the popular songs" use tons of the stuff. So, important? Without a doubt. 

But is it important to you, the player, while you're on stage connecting with your audience? And what on Earth is compression, anyway?

Compression (and it's close relative, limiting) essentially reduces dynamic range, so that the subtle, whisper-quiet things can be easily heard, and the loud things don't drown out everything else. 

You can see how this would be useful when you're recording, but what does it mean when you're playing your guitar or bass or what-have-you live?

Well, to me the most important thing is that it gives you some of the flavor of a cranked amp, but at any volume: you can get that edgy, ready-to-explode sound (and the all-important SUSTAINNNNNNNNNNNNNNNNNNN) without ruining everybody's hearing. 

 It's also very useful for extended techniques involving any kind of scraping, rubbing, blowing on the strings, etc, because the subtleties will not be lost: they will be loud and clear. All the audio fairies (overtones) get to frolic freely.

Compression is also VERY useful for nailing that Nashville sound. I remember reading an interview with a great Nashville session player (they are some of the best players in the world) who said an older master session player sat him down, handed him the then-new MXR Dyna-Comp and said simply "Son, use this on everything you do." Here it's more of a textural thing, using a lot of it so it's not subtle at all. And since they tend to play bright Fender guitars through bright Fender amps, the compression keeps the transients from becoming lethal. You can get away with a brighter signal because leaning into a high note on your Telecaster will no longer chip anyone's teeth.

Yet another Good Thing a compressor can do for you, the performer (the hits just keep on coming!): how about being able to jump from rhythm to solo without stepping on anything? Just turn on the compressor at the beginning of the song and leave it on. You can set it to make the single notes every bit as loud as a strummed chord.

  These are just a few of the reasons why the compressor may become the most useful thing on your pedalboard. If you don't own one, go to a music store and check one out! Spend a little time with it, adjust it in different ways, from subtle to extreme. You'll see what I mean.