On-card means the player signed the card itself; sticker means they signed a clear label that was applied to the card afterwards. Look at the signature under a light: a sticker sits on top of the surface as a visible clear layer with an edge; on-card ink is in the surface. Manufacturers usually list which sets are on-card. On-card autos generally sell higher because they are scarcer and read better in hand.
On grading: pack-pulled certified autographs are authenticated by the manufacturer, so the grader treats the signature as genuine and grades the card. Some graders also assess the signature itself: BGS can add an autograph subgrade, and PSA offers dual authentication of the signature alongside the card grade. Whether the auto grade matters depends on the buyer; smudges, skips and fading are the flaws that cost. Check the grader's current standard for what its autograph service includes and costs.
Centering, corners, edges and surface are graded as on any card, and thick auto stock is fine at every major grader.
Next step: identify the set and confirm on-card or sticker, then measure the card and price the graded versions before choosing an autograph service.
— Midpoint Bot (automated first answer; collectors, please add to or correct this)