Nobody chooses a rug for its edge. Everybody notices a bad one, usually about two years in, when it starts lifting at a corner.
Binding: tape, stitched through
A woven cotton or linen tape is folded over the edge and stitched through the backing. It is clean, quick, and available in any width and colour, which makes it the most flexible option in a scheme.
A wide binding in a deliberately contrasting tone is the cheapest way to make a plain rug look designed. Ask for mitred corners — gathered corners are the giveaway of a rushed job.
Serging: yarn, wrapped by hand
Yarn is wrapped tightly around the edge by hand so there is no tape at all, just a dense rope of colour that reads as part of the rug rather than an addition to it.
It costs more and takes longer, and it is the finish on most rugs that look expensive in a photograph. On a curved or round rug it is the better choice, because tape has to be eased around a curve and serging simply follows it.
Fringing: traditional, and a commitment
Fringe is the extension of the warp threads on a traditional woven rug, and on a modern tufted rug it is added deliberately for that look. It reads antique and it suits some designs beautifully.
It also needs combing straight after cleaning, and it is the part a vacuum eats. Choose it because you want it, not because it came with the picture.
Glued and heat-sealed edges: avoid
The cheapest finish bonds a tape on with adhesive or heat rather than stitching through. It looks identical on day one and it peels within a couple of years, usually taking a strip of pile with it.
If a quote does not say how the edge is attached, that is worth asking about before anything else on the specification.