Everything you wanted to know about Elliptic Curve Cryptography

So you've heard of Elliptic Curve Cryptography. Maybe you know it's supposed to be better than RSA. Maybe you know that all these cool new decentralized protocols use it. Maybe you've seen the landslide of acronyms that go along with it: ECC, ECDSA, ECDH, EdDSA, Ed25519, etc. Maybe you've seen some cool looking graphs but don't know how those translate to working cryptography. The articles you find online either don't answer your questions, or launch in to a 30 minute description of birational equivalence.

This is a companion discussion topic for the original entry at https://blog.fission.codes/everything-you-wanted-to-know-about-elliptic-curve-cryptography/