Grazing (also: "buzzing", "scratching") is the concept of moving the player character close to a bullet without actually hitting it. Depending on the game, this action rewards the player in a variety of ways; the general reward is an increase in player score.
The institution of grazing exists in order to encourage a more aggressive style of play by giving rewards for flirting with danger. While in some games it only exists as a minor scoring mechanism (Radiant Silvergun, Raiden Fighters series, Armed Police BatRider, etc.), some of the others base its scoring systems and sometimes even survival largely on this particular gameplay technique (Psyvariar series, Shikigami no Shiro series, Ibara Kuro: Black Label, and notably, Touhou series).
In the Touhou games, grazing bonuses can make up a sizable portion of one's score, depending on the particular game and, essentially, its execution. For instance, it is very prominent in Embodiment of Scarlet Devil, where the player's score factors graze points in the end-of-stage tallies. In Perfect Cherry Blossom, grazing is commonly encouraged during Supernatural Borders and boss spellcards, and is the key to high scores. Imperishable Night doesn't rely on grazing as heavily as the previous two games, so it's commonly used there as a means of milking certain boss patterns for points when the player is aiming for a really high score.
Mountain of Faith was the only recent Touhou shooter that didn't reward the player for grazing in any way whatsoever, much to the disappointment of the devoted fans. However, the technique came back in Subterranean Animism, where it's once again the primary source of high scores.
Grazing in Windows Touhou releases is usually accompanied by a sharp pop/crackle and a visualization depicting small white particles flying off from the hitbox area. Both are especially noticeable when scoring lots of graze points at once.
This technique is very effective and easy to take advantage of when combined with streaming.