I'm teaching myself some differential geometry in the hope to understand gauge theory properly. In the definition of the the pullback bundle I came across a strange notation that I've never seen before. The pullback bundle was defined, with $f: M \to N$, as
\begin{equation} f^\ast \mathcal{E} \equiv (f^\ast E, f^\ast \pi, M , F), \end{equation} where the total space is given by \begin{equation} f^\ast E \equiv N \times_M E := \{ (x,z) \in N \times E | f(x) = \pi(z) \}. \end{equation}
What does the subscript on the $\times$ operator mean? Is it something to do with the operation occurring on the manifold $M$?
This is an operation in that is sometimes referred to as "fiber product", and which has other names and a general formalization in category theory (see the comment of @Riquelme). I have never seen this particular notation for it, but it nonetheless makes a kind of sense.
The idea is that you are given two functions both with the same target $M$, in this case $f : N \to M$ and $\pi : E \to M$. You want to form a kind of restricted product of $N$ and $E$, which could be called the "product of $N$ with $E$ over $M$". This is a subset of the "true" product, and it's defined by the formula that you gave.
You can form a fiber product in many different categories: groups and topological spaces are two places where I see fiber products now and then (probably related to the fact that I am a geometric group theorist).
In your situation the fiber product is being put to work to form pullback bundles in various bundle categories, such as vector bundles or fiber bundles.