Pinching a plant simply means cutting off the tip of its stem, thus removing its growing point. It’s called pinching because you really can do it with two fingers, using your thumb and index (or the nails thereof if you have them!) like a clamp and thus cutting through the stem. You can also “pinch” with pruning shears.
Removing the growing point stimulates branching and is therefore used to get a fuller plant. Pinching will also, however, delay bloom, so use the technique in moderation, fairly early in the season.
Also pinching is used remove an unwanted flowers or flower buds at the tip of any stem. Coleus and basil, which tend to decline after they bloom, are often pinched to stimulate continuous foliage growth.