The four criteria
All four major graders score the same four attributes: centering, corners, edges, surface. The differences between graders are how those sub-grades roll up to a final number, how strict each step is, and what subgrades they publish on the slab.
Centering
Centering is the ratio of border on opposing sides. Left/right and top/bottom. A perfectly centered card reads 50/50, 50/50. PSA requires roughly 55/45 or better for a 10. BGS publishes the centering sub-grade on the slab. The unforgiving fact: a card with perfect corners, edges, and surface but 70/30 centering will not get a 10 from any major grader.
Corners
Look at all four corners under 10x magnification. Crisp 90-degree angles with no whitening, denting, or fraying. The most common corner damage is barely-visible whitening from sliding in and out of a sleeve over twenty-five years. PSA tolerates a tiny amount; BGS is harsher.
Edges
Run a fingernail very lightly along each edge. Whitening, chips, and roughness all knock the grade. Vintage Pokémon cards from the Wizards era had a soft cardstock that picks up edge whitening more easily than modern prints. This is why pack-fresh Base Set cards in PSA 10 are scarce despite huge print runs.
Surface
Scratches, print defects, indentations, holo scratches. The holo window on a Base Set Charizard is the most-scrutinized surface in the hobby. Hold the card at 45 degrees under a strong light and rotate it slowly; any hairlines or scratches will catch the light.
How the final grade lands
PSA does not publish a formal formula. The working understanding is that the four sub-grades are weighted roughly equally, with the worst sub-grade applying a slight drag on the final number. A card with three 10s and one 9 typically gets a final PSA 9. BGS uses a documented weighted average that gives less weight to the worst sub-grade, which is why "BGS 9.5" exists as a distinct grade and is more common than "BGS 10."
The PSA 10 vs PSA 9 question
For most cards the PSA 10 price is dramatically higher than the PSA 9. Charizard Base Set 1st Edition shows the most extreme version: roughly 20x. The practical implication is that the marginal cost of submission is small compared to the marginal value of a 10, but only if you have a real candidate. Submitting a card that is going to land at 8 or 9 on a $100 service fee is a losing trade unless the underlying is valuable enough.
PSA vs BGS vs CGC vs SGC
PSA leads the market for vintage Pokémon. PSA 10 1st Edition Charizards trade at a premium over any other slab. BGS dominates parts of the modern hobby and has the most-publicized subgrade tradition. CGC is the youngest of the four but gaining share, with very fast turnaround. SGC is rarer in Pokémon collecting but well-regarded in the broader card market. The cheap version of this: if you are getting the card graded to sell, PSA gets the best resale on Pokémon nine times out of ten.