I'll give one answer to get things started: discrete Morse theory.

A discrete Morse function assigns a real number to each face in a simplicial complex or more generally to each cell in a regular CW complex. (With care, one can also work with non-regular CW complexes.) While in Morse theory there are critical points, each having an index, the discrete Morse theoretic analogue is a critical cell, with the dimension of a critical cell playing the role of index of a critical point. The Morse inequalities still hold, and one can still calculate Euler characteristic as alternating sum of Morse numbers (i.e. alternating sum of the number of critical cells of each dimension). The original regular CW complex will be (simple) homotopy equivalent to a CW complex having fewer cells (unless all cells are critical), namely a CW complex whose cells are indexed by the critical cells.

This analogue with Morse theory was established by Robin Forman in his paper "Morse theory for cell complexes", Adv. Math., 134 (1998), no. 1, 90-145. Another nice reference is his paper "A user's guide to discrete Morse theory". The idea has proven quite useful in the study of various simplicial complexes e.g. in combinatorics, and the idea appeared independently in work of Ken Brown under the name "collapsing scheme".