ECC (elliptic curve crypto) build on elliptic curves (EC) .. not just any but the *discretized* version of a EC.

Further, in bitcoin and almost any other cryptocurrency (except NXT and CryptoNote) ONE particular curve is used.

That particular curve is the bases of all asymmetric crypto in crypto currencies .. so you have one public key and a different private key ..

in shares secret crypto systems, you just have one shared key (a password so to say)

the curve that is used in btc and here is called "Secp256k1"

https://en.bitcoin.it/wiki/Secp256k1and defined by a set of numbers:

The elliptic curve domain parameters over Fp associated with a Koblitz curve secp256k1 are specified by the sextuple T = (p,a,b,G,n,h)

these numbers basically come from 'somewhere' and in contrast to the SHA256 numbers we here have:

secp256k1 was almost never used before Bitcoin became popular, but it is now gaining in popularity due to its several nice properties. Most commonly-used curves have a random structure, but secp256k1 was constructed in a special non-random way which allows for especially efficient computation. As a result, it is often more than 30% faster than other curves if the implementation is sufficiently optimized. Also, unlike the popular NIST curves, secp256k1's constants were selected in a predictable way, which significantly reduces the possibility that the curve's creator inserted any sort of backdoor into the curve.

//edit:

BTW, the question around the "right" initialization values for hash functions and other crypto schemes has it's own wiki-page and covers a set of

"Nothing up the sleve" numbers:

http://en.wikipedia.org/wiki/Nothing_up_my_sleeve_number