If the clog is inside, it may take a little more work. Three things I have used include a gutter snake, (toilet snake), a hose end, and even a wire coat hanger. Each works best for a different type of clog.
A small garden hose can be used to unclog soft clogs that are cause by dirt and and leaves. Simple slid the end into the down spout and turn the hose on full blast. If the clog is easy, it will come right out and water will start flowing down the downspout again.
Difficult clogs will require a snake. Because gutter downspout are much smaller then toilets, you will want to use the thinnest snake you buy. Then snake the top until you get to the second turn in the gutter. Pull out whatever you can and the remainder of the clog will be flushed down.
Coat hangers, are a little more smaller then a snake and almost as strong. They are good at removing bits of wood that clog gutters on shake roofs. The one disadvantage is that they are often not very long. They work well for the first bend in the gutter but not very well for the second.
If you still can't unclog it, you may have to take the downspout of, clean it and then carefully reattach it to the house. When you do this, remove the nail on one side of each support bracket. Remove the downspout that is clogged. Turn it up side down and shake it or flush it with water. The clog in the down spout should then come right out. You can also look through it or tell by the weight whether it is still clogged. After you are done cleaning it, re attach it to the hose. Make sure the down spout is in the same location.