Taking an $\left[\left[n, k, d\right]\right]$ code:
The classical equivalent to this is an $\left[n, k, d\right]$ code, which is a code referring to the number of bits, $n$, encoding $k$ bits. The third number, $d$, is the minimum Hamming distance taken between any two codewords. This is equal to the minimum Hamming weight (i.e. number of non-zero bits) of non-zero codewords.
As per the classical case, in the quantum case, the first two numbers are referring to the numbers of qubits, $n$, that encode $k$ qubits. $d$ is still used to refer to distance, but the definition of distance has to be changed.
The weight, $t$, of a Pauli operator $E_a$, is the number of qubits that a (single-qubit) Pauli operator $\left(X, Y \text{ or } Z\right)$ acts on. As an example, arbitrarily taking $E_1 = X\otimes I\otimes I\otimes Z\otimes I$, $E_1$ has a weight $t=2$. The distance is then the minimum weight that the overlap of a Pauli operator (in the space of possible errors) acting on a codeword, with a different codeword is non-zero, or the minimum weight such that $\left<j\vert E_a\vert i\right>\neq C_a\delta_{ji}$ for some (real) $C_a$ for all codewords $i$ and $j$. That is, the distance is the minimum number of errors that can occur on a codeword that causes it to be mapped to a different codeword.
For more details, see e.g. Chapter 7 of Preskill's quantum computation notes.