Basically over time, micro contaminants will get stuck to your paint work. Stuff like rail dust, brake dust, or even overspray and etc.
If you've ever washed your car and wiped it clean and glided your hand across the surface of your paint, it may not feel smooth as glass and it may feel somewhat like ultra fine sandpaper. You may even hear it too.
A claybar looks and feels basically like....clay or blu-tack and you use a lube (or really anything like water is fine) and spray the surface if your paint and glide the claybar over it. At first you will feel resistance when you try and glide the claybar over the paint but go over it several times and it should glide very smoothly. The claybar picks up all the contaminants that regular washing cannot to.
After doing that you can wipe off the area and then give it another feel. Your paint will literally feel as smooth as glass.
It should be done after washing and before polishing and IMO before waxing too.
YouTube - Chrysler 300 clay bar video.
Heres a video of the process on youtube. That is a pretty extreme example as you can HEAR how bad it is literally...usually it won't be that bad but you can often actually hear it.
Extremely effective and extremely satisfying!!