a singular point
If an ordinary point is a safe place to expand, a singular point is a place to be wary of — a location where the equation's coefficients misbehave, and where the ordinary power-series method can break down. Singular points are not a nuisance to be avoided so much as a feature to be respected: the most interesting behaviour of a physical system — the spot where a vibrating membrane is pinned, the centre of an atom, the rim of a lens — often sits right at a singular point.
The definition is simply the failure of ordinariness. Put the equation in standard form y'' + P(x) y' + Q(x) y = 0. A point x0 is a singular point if P(x) or Q(x) is not analytic there — typically because one of them blows up at x0, which happens at the zeros of the original leading coefficient. For instance Bessel's equation x^2 y'' + x y' + (x^2 - nu^2) y = 0 has, after dividing by x^2, the coefficient P = 1/x and Q = (x^2 - nu^2)/x^2, both singular at x = 0; so x = 0 is a singular point of Bessel's equation, which is exactly the point physicists care about most.
Singular points come in two flavours that decide how hard the equation is to solve there: regular singular points, where the blow-up is mild and the method of Frobenius rescues you, and irregular singular points, where it is severe and even Frobenius may fail. Recognising that a point is singular is the signal to stop reaching for the ordinary recipe and to classify the singularity before going further.
Legendre's equation (1 - x^2) y'' - 2x y' + l(l+1) y = 0 has leading coefficient (1 - x^2), which vanishes at x = 1 and x = -1. Dividing, P = -2x/(1-x^2) blows up there, so x = 1 and x = -1 are singular points — the poles of the sphere, where the Legendre polynomials famously stay finite.
Singular points sit at the zeros of the leading coefficient — and those are usually the physically meaningful locations, like the poles of a sphere.
Not every singular point is a disaster: many are 'regular' and yield perfectly good (if more elaborate) solutions. The word 'singular' flags difficulty, not hopelessness.