Vibrato is that wobbly sound you often hear, in guitar solos especially, or when the player is playing melody lines, but you will also hear whole chords being 'vibrato-ed'. It's a technique that applies to most instruments and singing, the physics of which are to alter the pitch of the note in a pulsating manner. Singers do it with their voice boxes; we guitar players do it by altering the tension on the string once we've plucked a note. It's done to add expression and can even extend the sustain of the note as it dies away, but mostly we do it because it sounds neat. There are a couple of ways of doing it on a guitar.
The classical way is to move your finger back and forth along the line of the string between the fretwires wherever your fingertip happens to find itself. You need to be quite firm about it, and it's always going to be a very subtle effect, so don't expect great wide variations in pitch. I've always found that the action is more in the arm/wrist/hand that the fingertip itself. Don't think too much about the finger, concentrate on moving the weight of your forearm, but stay relaxed. Let gravity do the work.
The other way, by far the more common way on steel string and electric guitars, is to physically wobble the string sideways across the fretwire. You don't need much sideways movement, just enough to raise the pitch slightly, but you must allow the string to quickly relax back to its normal tension, then reapply the sideways pressure, and relax again. It's not the easiest technique to master, but once you do you will be able to alter the depth and speed of the vibrato at will. You will hear many players wait a second or two before applying it, letting the note sit still for a while, then wobbling it. Singers do the same with their voices, and it's a good idea to listen to singers and copy their timing ideas. Most players wind up with their own sound when it comes to vibrato. BB King, for example, has a very distinctive sound and way of playing vibrato. He's instantly recognizable once you hear that vibrato. He seems to lift everything but his fingertip off the finger board and flutters his hand like a butterfly. It's a very personal technique, involving microtones and nanoseconds, so make sure you experiment as much as you can with it to find your way of doing it.