What to do about clogged downspouts?
Disconnect the down spout.
This may require you to remove screws and/or rivets. Screws are easy, just use the appropriate driver and back the screw out. Rivets, however, will have to be drilled out. Select a drill bit just larger than the hole in the head of the rivet, place the tip of the bit in the hole and drill. You're not looking to drill straight through, you'll drill just enough so the head of the rivet comes off.

Inspect the drain
Now that you have access to the drain itself; using a flash light, inspect the opening looking for blockages. 

Clear the drain
You might get lucky here and be able to clear the drain by hand, or blow the clog out using a garden hose. If not, you'll want to snake the drain out using either a handheld or drill powered drain auger.



Reassemble
Once the drain is running freely, reattach the downspout using your preferred fastening method. 

Warning:
DO NOT use chemical drain cleaners. Some areas tie gutters into the storm sewer system, which may drain directly into nearby rivers and streams.