With the disclaimer that I am not a cellist, or indeed an expert... I can tell you what mutes do for violins/violas! (to my knowledge...

)
There are two basic sorts of mute, normal mutes such as one would use when it says "con sordino" (with mute) in music, which mute the sound, making it more hollow and with a very different colour.
These can (on violins at least) be made or wood, wire, metal, plastic or rubber.
"Normal" mutes I have seen include: wood or plastic ones which look very approximately like an m, where the 3 spokes of the M fit between the strings on the bridge; wire ones which look frankly a bit weird (I don't think I've ever seen one up close

), a wire contraption that again fits onto the bridge, and when not in use hangs on the strings on the tailpiece side of the bridge; and little round rubber ones that are circular (I've certainly seen these for cello) with a sort of niblet that fits over the bridge between the two middle strings, and again when not in use hangs on the strings behind the bridge.
Practice mutes are considerably heavier and bulkier, and not intended for use to actually play in orchestras or for solos. The one I have is a little metal weight attached to a metal clippy thing (again, vaguely m shaped) that attaches to the bridge, and deadens the sound to the point where you can hear it, but it's very quiet and not at all vibrant, and has little carrying power. I believe you can also buy ones made of heavy rubber.
Hope this helps/makes sense!