Hamming distance and parity
A (binary) code $C$ is a collection of binary strings of some length $n$, known as codewords. The minimal distance of $C$ is the minimum Hamming distance between (different) codewords. Hamming codes have minimum distance 3, which means that (1) every two codewords differ in at least 3 places, (2) there exist two codewords which differ in exactly 3 places.

Hamming codes are examples of an important class of codes known as linear codes. You can think of linear codes in many ways:


A linear code of length $n$ is a linear subspace of $\{0,1\}^n$. The dimension of the code is known as the rate $k$.
A linear code is a code which is closed under XOR, that is, the XOR of any two codewords is again a codeword. It turns out that the number of codewords is always a power of 2, say $2^k$. The parameter $k$ is known as the rate.
A linear code is a way of encoding $k$-bit messages using $n$ bits. The original bits of the message are put in fixed places, and every other bit is an XOR of a certain fixed subset of the message bits. For example, the SEC(7,4) code encodes a 4-bit message $x,y,z,w$ using 7 bits:
$$
x \oplus y \oplus w, x \oplus z \oplus w, x, y \oplus z \oplus w, y, z, w
$$


The minimal distance of a linear code is also the minimal Hamming weight of a non-zero codeword. So the fact that Hamming codes have minimum distance 3 means that (1) every non-zero codeword has at least 3 ones, (2) there exists a codeword having exactly 3 ones.

We have mentioned three parameters: the length of the code $n$, the rate of the code $k$, and the minimal distance $d$. There is no exact relation between these three parameters, though there are various  bounds relating the three quantities. One of these bounds is the Hamming bound:
$$
\sum_{\ell=0}^{\lfloor \frac{d-1}{2}\rfloor} \binom{n}{\ell}\leq 2^{n-k}.
$$
For example, for the SEC(7,4) code this gives
$$
\binom{7}{0} + \binom{7}{1} \leq 2^{7-4}.
$$
You can check that equality holds, and indeed this is the case for all Hamming codes. Codes for which equality holds are called perfect codes. Apart from a few trivial and exceptional cases, Hamming codes are (essentially) the only perfect codes.