Following Bar-Natan's down-to-earth approach, I will explain Khovanov's construction which associates to a knot a certain complex of graded vector spaces. If two knots are ambient isotopic their complexes are homotopy equivalent (grading is preserved). Therefore the cohomology groups of the knot complex are knot-invariants. It turns out that the Jones polynomial of a knot equals the graded Euler characteristic of the knot cohomology. Khovanov derived a more general polynomial from his complexes which is a more powerful knot invariant, as has been shown by explicit computations. References:

1. Dror Bar-Natan, "On Khovanov's categorification of the Jones polynomial", Algebraic and Geometric Topology 2 (2002) 337-370; math.QA/0201043.
2. Mikhail Khovanov, "A functor-valued invariant of tangles", Algebr. Geom. Topol. 2 (2002) 665-741; math.QA/0103190.
3. Mikhail Khovanov, "A categorification of the Jones polynomial", math.QA/9908171.