What is the meaning of the subscript sometimes seen after Cartesian products in geometry? I often seen a subscript after a Cartesian product, e.g. $A \times_\kappa B$, for example in this Wikipedia article about the spin bundle. What is the meaning of this generalised Cartesian product? Thanks.

This usually denotes a fibered product or pullback, a categorial generalisation of the ordinary product. 
As you tagged (general-topology), I will describe the pullback in the category of topological spaces, but let's begin with the categorical description: Suppose we are given three spaces $X$, $Y$ and $Z$ and two continuous maps, $f \colon X \to Z$ and $g \colon Y \to Z$, think of them as follows: 
$$  
  \begin{matrix}
       && X\\
       & &\downarrow f\\
      Y & \stackrel g\to & Z 
   \end{matrix}
$$ 
Then, the pullback $X \times_Z Y$ (along $f$ and $g$) is given by a topological space $X \times_Z Y$ and to continuous maps (called projections), $\pi_X \colon \def\p{X \times_Z Y}\p \to X$ and $\pi_Y \colon \p\to Y$, such that 
$$  
  \begin{matrix}
     \p  &\stackrel{\pi_X}\to & X\\
      \downarrow \pi_Y& &\downarrow f\\
      Y & \stackrel g\to & Z 
   \end{matrix}
$$ 
commutes and $\p$ is universal with this property, that is if we are given a space $P$ and maps $p_X \colon P \to X$ and $p_Y \colon P \to Y$, such that 
$$  
  \begin{matrix}
     P  &\stackrel{p_X}\to & X\\
      \downarrow p_Y& &\downarrow f\\
      Y & \stackrel g\to & Z 
   \end{matrix} \tag 1
$$ 
commutes, there is an unique continuous $i \colon P \to \p$ such that $\pi_X i = p_X$ and $\pi_Y i = p_Y$.
For topological spaces, the pullback is given by the subspace (with the subspace topology) of the ordinary product 
$$ X \times_Z Y := \{(x,y) \in X \times Y: f(x) = g(y) \} $$
and the projections are the usually coordinate projections (to be more exact, their restrictions). For suppose, we are given some $P$ as in (1), we may define $i \colon P \to \p$ by $i(q) = \bigl(p_X(q), p_Y(q)\bigr)$, as $p_X$ and $p_Y$ is continuous, $i$ is, as it is well-defined, since (1) commutes. $i$ is unique, since we must have $\pi_X i = p_X$ and dito for $Y$. 
