What does 63 stand for when describing the caliber of 7.62×63mm bullet?

63 refers to the length of the cartridge case, and 7.62x63mm is an American Springfield.30-06 rifle bullet, which was improved from.30-03 rifle bullet (and.30-03 was developed together with M 1903 rifle).

30 is 0.30 inch (7.62 mm) and 03 is 1903. 30-03 is a round-headed bullet, which was improved into a pointed bullet in 1906 and named as 30-06 rifle bullet. The length of the cartridge case is 63.35 mm and the total length of the bullet is 84.84 mm.

7.62x39mm is an AK-47 bullet, with a shell length of 39mm, a warhead length of 26mm and a total length of 55.7mm.