Propagation is just persuading a piece of a plant to grow its own roots. Which method works depends entirely on how the plant grows. That is why the per-species guides below matter more than any general advice.
The four methods that cover most houseplants
Stem cuttings
Cut below a node and root the piece in water, or put it straight into moist mix. Works for anything that climbs or vines: Monstera, Philodendron, Pothos.
Division
Unpot the plant, separate the root ball into clumps, and repot each one. This is the only option for plants that grow from a central crown rather than a stem.
Offsets and corms
Some plants produce miniature versions of themselves attached to the parent. Alocasia grow corms along their roots. Separate them and pot them individually.
Leaf cuttings
A single leaf regenerates a whole plant. This is limited to a handful of genera: Begonia, Sansevieria and most succulents.
Why cuttings rot instead of rooting
Rot is almost always one of three things: no node included in the cutting, water that was never changed, or a cutting taken from a stressed parent plant. A cutting has no roots and cannot defend itself. It needs clean water, warmth and light while it works.