We normally differentiate between Cloths and Ridgid discs. The former being specialised textiles laminated to make either a self-adhesive cloth or a magnetic cloth.
These are by far the most common ways of polishing. A cloth has quite a bit of room for accepting abrasives and lubricants which in turn makes it easy to use, not needed a precise and contant flow of suspension and lubricant to perform well. The roomy texture is on the other hand the reason for the typical edge rounding associated to cloths. This is why the latest additions to polishing has been the ridgid discs.
These are polymer composite discs coated onto a steel disc for magnetic adaptation. The composite surface is generally harder than textile and will therefore give excellent edge retention. But the composite surface also allows a higher degree of contact between abrasive and sample surface thus allowing for higher abrasion resulting in quick preparation. The latest addition to this line is Allegran-3 and Allegran-1. The combination of high removal and fine surface finish often allows easy 3 step methods.
Magnetic discs are by far the easiest way of holding consuambles on the disc. Some consumables might not be available in a magnetic form, thus an adapter is necessary.