Contents
What Is A Signature?
First, let's define what a signature is. In it's most basic meaning, it's a proof of identity. To sign something is equivalent to saying: "These scribbles represent me, and I agree with, stand by, or verify whatever is written in this piece of paper."
There are a few problems with analog signatures, though:

It's trivial to copy the signature. When you sign a document, you are in effect making your "private key" public. Anyone can lift it and sign something else without your knowledge.

The signature is not bound to the contents of the document. The document can be modified after you sign it, and the signature would not reflect that.
Making It Digital
Digital Signatures make use of asymmetric cryptography, i.e. a public and a private key. In the ECDSA scheme, the signature is an aggregate of the private key, a nonce, and a hash of the document. It is represented as a pair of numbers: (r, s). Then, you can verify the authenticity of a signature with the document, the corresponding public key, and (r, s).
This allows for the following properties:

The signature is tied to what was signed. Changing a single bit will make the signature verification fail.

The private key is not made public when you sign something, making forgery impossible (if you don't reuse the nonce, that is. This is how the PlayStation 3 signing key was extracted).
But how does it actually work?
The Math Behind It
Let's say Alice wants to sign a contract with Bob. Bob needs to know that: (a) it was actually Alice that signed it, and (b) that Alice did not sign a modified contract.
Note: each elliptic curve will generate a different signature. In this case, we're using secp256k1. I wrote about it in more detail here.
Sign
Let z be the hash of this contract, k be a random number, e be a private key, n the order of the finite field, and G be the generator point for the curve.
To create a signature, we have to derive r and s:
r is the x coordinate of the public key derived from k:
R = kG
r = R_{x} mod n
Note: r must be different from 0. If this is the case, use another nonce.
s incorporates the private key e, the hash z, the nonce k and r:
s = (z + re)/k mod n
Note: s must be different from 0. If this is the case, use another nonce and start from the beggining.
You have created a signature (r, s).
Verify
Now, to verify a signature, follow this process:
Let z be the hash of the contract, (r, s) the signature, and P the corresponding public key.
Then, let u = (z/s) mod n and v = (r/s) mod n (this is why r and s cannot be 0).
Compute uG + vP = R. If r is equal to R_{x}, then the signature is valid.
I've summarized all of this in this gist: