The term c-number is used informally in the way Meer Ashwinkumar describes. As far as I know, it doesn't have a widely promulgated formal definition. However, there is a formal definition for c-number that agrees with the way the term is used in many cases, including the case you're asking about.
As you may know, you can think of the operator formalism for quantum mechanics as a generalized version of probability theory, in which real-valued random variables are represented by self-adjoint operators on a Hilbert space. More generally, complex-valued random variables are represented by normal operators.
A c-number is a random variable represented by a scalar multiple of the identity operator.
Intuitively, a c-number is a random variable that isn't really random: its value is a constant. The identity operator itself, for instance, represents the random variable whose value is always $1$, while $-4$ times the identity represents the random variable whose value is always $-4$. You can see why this makes sense by computing the expectation value, variance, and higher moments of a c-number relative to some state.
In your example, Tong is talking about a model for a random scalar field,^ whose amplitude at the point $x$ is the real-valued random variable $\phi(x)$. For any two points $x$ and $y$, the commutator $[\phi(x), \phi(y)]$ represents an imaginary-valued random variable. The commutator turns out to be a multiple of the identity—in other words, a c-number. Since this c-number depends on $x$ and $y$, Tong calls it a c-number function (of $x$ and $y$).
^ A free scalar field can be seen as a quantum version of white noise.