I use the Meguire's clay when I really want to "go to town" on my car.
If you've cleaned your car, and think it's an awesome job, try gently running your fingernails over the paint.
Feel all those little gritty bumps as you do it? That's all of the contaminants that no amount of washing will remove - they've practically "bonded" to your paint.
The clay simply allows you to remove these "bonded" contaminants from the paint, leaving it with that "smooth as glass" feel.
It takes a fair while, but if you see doing this as part of a "major service" of your paintwork (i.e. do it every 4-6 months) contaminants will find it harder to bond to your paint, and your car will stay clean looking for longer
I usually:
1) Wash and chamois the car (changing the water frequently, and don't dip your sponge in the bottom 3rd of the water in the bucket - that's where all the dirt is, and you'll simply be rubbing this all over your duco if you do.
2) Use a paint cleaner to prepare the surface (I use Meguire's paint cleaner)
3) Do the "fingernail test" if the test returns "positive" - clay away, one panel at a time until you get that smooth as glass feel. If you notice the clay is getting dirty, just knead it until it disappears and continue.
4) Polish the car with a high quality polish - the most common mistake is too use too much, but in fact being quite stingy returns the best results. Also, never pour the polish directly onto the paintwork - always pour it onto an applicator pad first, then apply that to the paintwork using small, circular strokes, working on one section at a time.
5) Buff off, using only a 100% cotton terry toweling cloth. If you get any sort of residue build up on the cloth, change to a clean one.
6) Apply a high quality carnuba wax to the entire paintwork, i.e. Mequire's, Zymol, etc as per steps 4 & 5. This forms a hard exterior shell for the paintwork, that protects the cleaned, polished surface from contaminants (i.e. it's harder for contaminants to bond to the paintwork), and stops it getting dirty as quickly , and makes it much easier to clean for the next few washes if done properly.