Something that's clogged is blocked or stopped up. When your sink is clogged, the soapy water can't flow down the drain.
If your gutters are clogged with leaves, rain water will simply flow and drip over the edges, and if your arteries become clogged, you'll be at a higher risk for having a heart attack. In both cases, the clogged things need to be cleared or cleaned out in order to work properly. Another way something can be clogged is to be clotted or thickened: "I don't like this oatmeal because the pieces are all clogged together."