a blow-up
Imagine standing at a point where a curve crosses itself and wanting to pull the two branches apart so you can see them separately. A blow-up does exactly this: it replaces a single point by all the directions through it (a whole projective space of tangent lines), separating things that were tangled together at that point. It is the basic surgery for resolving singularities — turning a pinched or self-crossing variety into a smooth one without changing it anywhere else.
The blow-up of affine space A^n at the origin is the variety Bl_0 A^n = { (x, [l]) in A^n x P^{n-1} : x lies on the line l }, together with the projection pi back to A^n. Away from the origin pi is an isomorphism, because a nonzero point determines its direction uniquely; but the entire fiber over the origin is a copy of P^{n-1}, the exceptional divisor E, recording every direction through 0. To blow up a variety X at a point, take the closure inside Bl_0 A^n of the part of X away from the origin; this gives the strict transform of X. The effect on a singular curve is to separate branches according to their tangent directions: a node, whose two branches have distinct tangents, is resolved in a single blow-up because the branches land at different points of E.
Blow-ups are the engine of resolution of singularities: Hironaka's theorem says that over a field of characteristic zero, any variety can be made smooth by a finite sequence of blow-ups along smooth centers, and they are a basic tool in birational geometry, since a blow-up is always a birational morphism (an isomorphism over a dense open set). Caveats worth keeping straight. A blow-up does change the variety: it is birational, not an isomorphism, and it raises some cohomology and lowers self-intersection numbers. Not every singularity resolves in one step — a cusp, whose two branches share a tangent direction, needs more than one blow-up — and in positive characteristic resolution is far subtler and only recently established in low dimensions.
Blowing up A^2 at the origin and taking the strict transform of the node y^2 = x^2(x + 1): in the chart y = tx the equation becomes t^2 = x + 1 after dividing out x^2, a smooth curve meeting the exceptional line in two points t = 1 and t = -1. The single self-crossing has been pulled apart into two separate smooth points.
A single blow-up resolves a node by separating its two distinct tangent directions on the exceptional divisor.
A blow-up is birational but not an isomorphism — it replaces a point by an exceptional divisor and changes cohomology. A cusp (branches with a shared tangent) does not resolve in one blow-up; several are needed.