Somewhere along the way, an unknown person discovered that running a slide made of metal, glass or bone along a guitar string would produce a sound very similar to the dynamics of a the human voice. It was an expressive way to compliment the singing and produced tones and notes that were otherwise impossible.
When playing slide guitar, you lay a piece of something hard on the strings and slide it around the neck. Most people use a steel or glass cylinder, the neck of a wine bottle is excellent.
The important thing is not pressing the strings down to the fingerboard like you do when playing the guitar normal style. Just let the slide rest lightly on the strings, this produces a very smooth and whining sound, and you can hit any note -- not just the ones between the frets.
Since you can play almost any note on your guitar using the slide, it usually takes some practice to hit the right note -- it feels more or less like playing the violin (with a piece of glass stuck on your finger).
You should play slide on either a steel string acoustic guitar, or an electric guitar. Classical style guitars with nylon strings won't make enough sound, although the tone they produce tend to be very warm and soft when played using a slide. An electric guitar with a little bit of overdrive is king!
Most slide players use open tunings when sliding because you, practically speaking, move a barre chord up and down the neck, sometimes throwing in a fretted note or two. Open tunings allow you play a "real" chord with a barre. The most widely used tunings are open G and open E.
You should pick up recordings by these: