Clogged corner
Water flows to the corner of the this gutter rather than out the drain pipe. This corner is clogged because of dead leaves that are in a pile on top of the roof which causes water to fly off the roof.
Water Damage
Because the water isn't flowing out the drain properly it is flying off the roof and over on to the side of the house continuously as it rains.
Clogged pipe
The drainage pipe in this picture is clogged, otherwise water would not be pouring out the the side.
Leaves in the Gutters
Leaves in the gutters clog up it up making it hard for water to pass through and get out successfully.
Plant Growth
Over time if the gutters aren't cleaned out like they should be, nature will take over and start sprouting plants inside them.
Nice Warm Home
Leaves can help make a nice home for critters looking to settle in. Mold and insects like nice damp places. If you are wanting to avoid these potential problems, then clean out those gutters.