An EC (Elliptic Curve) key-pair is a pair of a private and public key constructed from a given subgroup generator in a given elliptic curve group.
Here are the steps to generate an EC private and public key pair:
EC Private and Public Key Pair
Naming Elliptic Curves Used in Cryptography
There is an infinite number of elliptic curves, but a small number are used in cryptography, and these special curves have names. Apparently, there are no hard and fast rules for how the names are chosen, but there are patterns.
The named elliptic curves are over a prime field, i.e. a finite field, with a prime number of elements p. The number of points on the elliptic curve is on the order of p [1].