Playing a guitar without vibrato is like eating Mexican food without salsa. Vibrato is one of the most definitive techniques in defining your musical style, your own personal unique sound. No two players have the same vibrato sound. Some play slow and wide while others play narrow and fast, and thats the beauty behind spending a lot of time learning different vibrato techniques. Its like eating different salsas every time you go to your favorite taco stand. Enough talk about food, lets talk technique.
Vibrato is sometimes confused with tremolo. Basically it is just a technique of vibrating the string sharp and flat around a root note. For example if you are playing the 7th fret on the 3rd string with your 3rd finger on your fret hand you can just push the note up slightly and them pull it down slightly. Do this in quick succession so the over riding effect is a wobbly tone around the root note you are playing. Use your hand, wrist, and arm to make the movements. Never just wiggle a finger. Some people use a technique of pulling the string down towards the floor and then releasing back up.
Try this in a variety of speeds and levels of bending during your vibrato....