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Resolution of singularity of curves (over algebraically closed fields). A somewhat long elementary proof is given e.g. in chapter 7 of Fulton's Algebraic Curves (which he made available online for free). Basically you use the primitive element theorem to reduce it to the planar case. And then for a planar curve you track the changes in multiplicity of a point under blow-ups via explicit computation. A more abstract approach, given for example in chapter I.6 of Hartshorne's, identifies a nonsingular curve with the set of discrete valuations of a function field, and shows that any projective morphism from an open subset of a nonsingular curve can be extended to the whole curve.