If your gutter is overflowing but there is no obvious blockage along the length of the guttering, your downpipe might be blocked. There are several methods you can use to unblock a downpipe, depending on the type of pipe you have and where the blockage is located (top, middle or at the bottom).
Blocked at the Top
Rig up a ladder safely and have a look down inside the blocked pipe. If you can see the blockage you can try to hook it out with a length of wire with the end bent into a hook shape. Before you start to unblock the pipe, place a piece of wood over the drain at the bottom of the pipe to prevent any loosened material falling down the pipe and potentially blocking the drain as well.
If you cannot hook the blockage out, work the wire up and down through the blockage until it starts to break apart and fall away. You should then pour a couple of buckets of water down the pipe to flush out any remaining material.
Blocked in the Middle
A blockage in the middle of a downpipe, especially in a swan-necked downpipe, is more difficult to remove. You can hire flexible drain clearing rods to help do the job or you can try dismantling the downpipe. The trouble with dismantling the downpipe is that it can be difficult to know where the blockage actually is and so you might need to remove almost all of the pipe. You can try tapping the pipe at intervals along its length and listening for a more solid sound. This should at least give you some idea of where the blockage is.
To dismantle the downpipe, start at the bottom by removing the two screws that hold the lowest pipe clip to the wall. If the pipe clips are the two-part type, leave the backplate in place and just remove the pipe ring. Carefully remove the loose section of downpipe (gentle force should pull it out of the section above it) and have a look inside for the blockage. Continue to remove clips and pipe sections until you find the blocked section. When you find the blocked section, use a length of wire or bamboo to dislodge the blockage.
Check to make sure that there isn’t a second blockage further up the pipe by pouring a little water down from the top. You can then replace the sections of pipe and reattach the clips to the wall as you work your way down.
You can buy wire mesh or plastic caps to fit into the top of most types of doownpipe which allow the water through but trap any leaves or dirt. Make sure that any cap you choose fits tightly and will not slip down the pipe, as this will obviously cause a blockage rather than stop one.
If your gutter feeds into a hopper at the top of a downpipe (often found in older houses), securely fix a piece of galvanised chicken wire or similar mesh over the top of the hopper and secure it in place with thin galvanised wire.
Consider cutting back any trees which overhang the house and the gutter to help prevent future blockages. If the tree is not in your garden, make sure you get permission before you start lopping branches off of it. Also make sure that when trimming branches, they do not fall and damage the gutter or, even worse, the roof.
If your downpipes are cast-iron, preventing blockages is even more important. A blockage made of leaves, etc, could freeze and expand in cold weather and possibly crack the downpipe. There is little you can do for a badly cracked cast-iron downpipe other than replace it.