Incorporating vibrato into a solo is something that usually comes naturally after years of playing. I never practiced to add this technique to my playing, it's something I realized I was doing when someone asked me "how do you do that wobbling of the string thing?" as I was playing.
The string is vibrated by bending and releasing rapidly by the fretting (left) hand. You can also use the tremolo bar.
If you vibrate the string using the fretting hand technique, you raise the note on the bend then lower the pitch on the release. When you use the tremolo bar technique, you lower the note when you push the trem' bar down and raise it when you release it. These two vibrato effects are subtly different, generally you use the fretting hand technique to vibrate the note.
Watch the video below to see exactly how to play this effect.
Vibrato is a quick bend & release technique, usually in time with the song you are playing. Push the string up, then release, then pull the string down & then release again. Do this in sixteenth notes (4 every beat). It doesn't have to be in time with the song but it does make it sound better.
Wide or Exaggerated Vibrato
The pitch is varied to a greater amount by using the fretting (left) hand or tremolo bar.
Adding vibrato into a solo is usually done when there's a note being held for a long period of time. It make the note or solo sound more professional and also sustains the sound longer. When a guitar is distorted you can hold the note indefinitely with a good vibrato technique.