Your "arteries" are blood vessels which carry blood away from your heart within your body. In addition to arteries, you also have "veins" which carry blood back toward your heart.
When an artery gets "clogged", it means that something is blocking up the artery and there's a high risk that you could have a heart attack.
English speakers generally believe that eating fatty foods "clogs your arteries" and causes heart attacks. Even though veins can get clogged as well, we always use "arteries" in this expression.