It's kind of a complicated topic. When I say separable I actually mean R-separable, which is more complicated again. I'm not sure how much calculus you know, but I'll have a go at explaining separable without the R-.

Laplace's equation in 3D is a partial differential equation with infinitely many solutions. The solutions are scalar functions f(x,y,z). The initial conditions restrict things so that only one solution is possible. An example of an initial condition might be "The solution is equal to a constant everywhere on this sphere". The shape of the condition tells you what coordinate system to use, so with this example we'd use spherical coordinates.

Let's say we want to solve Laplace's equation in Cartesian coordinates. We look for solutions of the form

f(x,y,z) = X(x) Y(y) Z(z).

So we have a product of functions of only one variable each. This is called

separation of variables. This breaks Laplace's equation into three separate "ordinary differential equations", which you can solve with second or third year calculus techniques (or in practice, with Mathematica).

Now let's say the initial conditions have spherical symmetry, so we convert to

spherical coordinates. This transforms Laplace's equation into a totally different equation involving r, θ, φ instead of x, y, z. Now we solve this by supposing the solution is a product of three functions of these variables.

f(r, θ, φ) = R(r) Θ(θ) Φ(φ)

Again, this breaks down the problem into three easier problems.

Sometimes if a coordinate system is too complicated, this method doesn't break down the system sufficiently, and you can't get a solution. We say that the Laplace equation is

separable in a coordinate system if separation of variables reduces the equation to three (or n, in our case) ordinary differential equations.