Tack cloth is a loosely woven fabric embedded with wax. It is used before painting or staining to remove even the finest dust that might otherwise be missed by wiping with a normal cloth or by vacuuming the object. Its natural stickiness picks up and removes dust without water, which is very helpful since water can take some time to dry, or can cause wood to swell which makes the surface no longer smooth and flat. Because it is tacky, you will probably want to wear gloves so your fingers don't get sticky or have residue. [It's completely safe and non-toxic; it just might be a little annoying to have residue on your hands.]
Tack cloth cannot be washed out and reused, so it should be thrown away when it is full of dust or debris. You can cut down larger pieces of tack cloth into smaller pieces if desired, but be aware that doing so may leave residue on your scissors, so don't use good scissors for this purpose.
When using tack cloth, first vacuum or clean the surface with a clean, dry rag, to remove the larger particles, then follow up with the tack cloth to remove the smallest dust. Use only light or no pressure -- little pressure is needed because the dust will be attracted to the tack cloth, and applying heavy pressure can actually rub off some of the wax onto your surface, making it no longer smooth.