A pinch harmonic (also squealy) is a technique to achieve artificial harmonics in which the player's thumb (on the picking hand) slightly catches the string after it is picked.
Most people would stop here, but I would add to the description above that the harmonic is enhanced and sustained with the help of the fretting hand by applying vibrato to the pinched note.
Let me detail the technique for you in 3 simple and easy-to-follow steps:
1) Let's consider the strings to be the initial position of 0 degrees.
When picking, you would normally have your pick be parallel or close to parallel with the strings. Therefore, your pick is also adopting the 0-degree position described above.
2) When you want to perform a pinch harmonic, you need to change your pick direction to form a 45 degrees angle (facing the floor) with the string. This is achieved by twisting your picking hand a little bit (just like shown in the video lesson).
3) Pick the string with a downstroke.
Just as you pick the string, your picking hand thumb should slightly touch the vibrating string. In other words, this happens directly after picking the string.
Your thumb catching the string will cancel (silence) the fundamental frequency of the string, and only letting one of the overtones dominate.
For better representation, watch the video lesson.
Apply vibrato, using the fretting hand to give the note much more sustain and room for squealing.