Aquarium moss is the plant you buy once. Like its land cousins it regrows readily from fragments, so a golf-ball-sized portion becomes as much as you want with a little time. Here is how to multiply it on purpose.
The basic principle
Every trim is propagation. Cut a clump of Java moss, Christmas moss or the like and each fragment can grow into a new plant, so the clippings are not waste, they are your next batch. The two things that decide success are spreading it thin and keeping it clean. There is no seed and no rooting to wait for; a broken strand simply carries on growing from where it was cut.
Spread it thin
The commonest mistake is piling moss on in a thick wad. The inside of a thick clump gets no light or water flow, browns and rots, while only the surface grows. Tease your moss into a thin layer over wood, rock or mesh, tie or glue it down lightly (cotton thread or a dab of cyanoacrylate gel), and it spreads outward into a clean carpet far faster than a heap ever would. See aquarium mosses for attaching, and the species comparison for how the different mosses behave.
The mesh mat method
For a moss carpet or wall, sandwich a thin scatter of fragments between two pieces of plastic mesh, or tie it to a single flat mesh panel, and lay it where you want cover. The moss grows through the mesh and knits into an even mat you can lift, trim or split. Trimmings from the mat start the next one, so one panel quickly becomes several.
Emersed and dry-start growing
Moss often bulks up faster grown emersed, that is out of the water but in saturated, humid air, than fully submerged. Lay fragments on a damp surface in a covered, humid container with bright indirect light, much like a terrarium, and let it carpet up before transferring it to the tank. Aquascapers use the same idea as a dry start: plant the moss on damp hardscape in a sealed, misted tank, grow it in emersed for a few weeks, then flood, as set out in the dry start method. It is the quickest way to a dense carpet.
Common setbacks
Two things spoil most attempts. The first is browning at the core of a clump left too thick, which is not disease but simply starvation of light and flow; the cure is always to thin and re-tie. The second is algae, which loves the same light the moss does and threads itself through the strands until the moss suffocates. Keep nutrients in check, do not overfeed the tank, give gentle flow, and lean on grazers; a small colony of shrimp earns its place picking detritus out of the moss, as covered in moss for shrimp. If a patch turns brown, do not bin it: trim away the worst, thin what remains, and the healthy fragments will regrow.
Keep it clean and lit
Moss grows faster with decent light and gentle flow, and CO2 helps if you run it, but the bigger issue is cleanliness. Trim to shape whenever it gets shaggy, and pass the clippings to another fishkeeper; moss is the friendly currency of the hobby. For why it is slow at first, see how fast does moss grow.