How do you derive the public key from you random 256-bits integer?
The curve's base point is a well-known constant. Since Bitcoin uses SECp256k1, the base point is:

04 (uncompressed point)

79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9
59F2815B 16F81798 (first coordinate)

483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448
A6855419 9C47D08F FB10D4B8 (second coordinate)

(See page 15 of my link.)

The private key is an integer from which you can derive the public key and produce signatures that can be verified with the public key.