The equation in question is P(A|B) = P(B|A) P(A) / P(B). In modern Kalman filter literature, this would be stated as something like: P(x_k | z_k) = P(z_k | x_k) P(x_k) / P(z_k). It is generally left as implicit in these sorts of equations that everything is also conditioned on the sequence z_1 to z_{k-1}. In this equation, x_k is a free variable in the state space (possibly multi-dimensional, so a vector), while z_k is the measurement, which is a realization of the random variable distributed as N(H(x_k), R_k). The result is a PDF over the free variable x_k.
So let's tackle the terms one by one:
1. P(z_k | x_k) - this is the probability that we measured z_k, given that the true object is at x_k. This is the aforementioned normal distribution N(H(x_k), R_k).
2. P(x_k) - this is the prior probability of the state estimate, generally after propagation through the motion model from P(x_{k-1} | x_{k-1}). In the Kalman filter, this is also Gaussian, and conveniently has the same mean as the term above (see note below).
3. P(z_k) - this is the denominator that someone else mentioned earlier can be effectively ignored, which is right - you only need it to normalize the numerator. If you must compute it, it can be factored as the integral over the entire state space of P(z_k|x_k)*p(x_k). Given z_k, this is a number, not a function.
4. P(x_k | z_k) - The result, which is a Gaussian PDF. You can arrive at it numerically by plugging in specific values for x_k, in which case (1) and (2) are numbers. Or symbolically, in which case (1) and (2) are functions, and you'll end up with the form of a Gaussian PDF.
Note: The original article quotes a distribution for the product of two Gaussians with arbitrary means. It does not state that this is an approximation, which is exact only in the case of equal means. This is why unbiased measurements are one of the Kalman assumptions.