pole
A pole is the tamest way for a holomorphic function to blow up: at a pole the function rockets off to infinity, but in a controlled, predictable manner, like a negative power of the distance to the bad point. As you approach a pole, |f(z)| -> infinity smoothly and the function behaves essentially like 1/(z - z0)^m for some positive whole number m. Poles are the well-understood singularities — civilized infinities, in contrast to the chaos of essential singularities.
Precisely, an isolated singularity z0 is a pole of order m (a positive integer) if, near z0, f(z) can be written as g(z) / (z - z0)^m where g is holomorphic and nonzero at z0. Equivalently, the Laurent series of f about z0 has finitely many negative-power terms, the most negative being (z - z0)^{-m}. A pole of order 1 is called simple. The defining behavior is that |f(z)| tends to infinity as z -> z0, which distinguishes a pole both from a removable singularity (f stays bounded) and from an essential one (no limit, finite or infinite, exists).
Functions whose only singularities are poles are called meromorphic, and they form a beautifully behaved class — ratios of holomorphic functions, like rational functions, are the prototype. At a pole of order m, the residue and the principal part of the Laurent series can be computed by a clean derivative formula, making poles the bread and butter of residue calculus. One caution: 'order m' counts multiplicity, so a double pole (m = 2) blows up faster than a simple pole.
f(z) = 1/(z - 2)^3 has a pole of order 3 at z = 2; as z -> 2 the modulus blows up like 1/|z-2|^3. The function 1/sin(z) has simple poles at every integer multiple of pi, since sin has simple zeros there. By contrast sin(z)/z has no pole at 0 — its singularity there is removable.
Order counts how fast the function blows up.