I would use either SDR 35 or Schedule 40 PVC under the pavers. The SDR 35 is the green stuff stxkyboy refers to, it is rated as drain/sewer pipe, and is slightly cheaper than sch 40.
Once the pipe was past the pavers, depending on where you were going to place the outlet, I would consider adapting the solid PVC to corrogated to finish the run. The reason for that is the corrugated is more flexible and forgiving if the trench isn't absolutely straight.
Make sure you have fall from the end of the patio to wherever you outlet the drain, you don't want the water to run back and collect on the pavers.
Ideally, you would want a fair amount of fall from the downspout to the outlet, somewhere in the range of 1/4" to 1/2" of fall per foot of pipe run.
If there is a ditch nearby that the water could be outletted to, that would be best. Otherwise, if you end the run with corrogated, use a Tuff-tite box (or similiar) at the end of the run to allow the water to outlet from. If the end of the run is with solid PVC, you may be able to use the Tuff-tite, but you can also check into one of those pop-up emitters. I've never used one of those, so I don't know if/how much they might restrict flow.
Hope it helps.
Dan