Why does a Capacitor allow AC but not DC?

A capacitor is two plates next to each other with a gap in between.

When you hook it up to a current, the plates will begin to accumulate charge until they are "full". This happens fairly quickly, but not instantaneously. If you're using DC current, the current will stop when the capacitor is fully charged. Current can't flow in between the gap of the plates.

AC current switches polarity continuously. It does this faster than it takes the capacitor to fully charge. Before the capacitor is "full", the polarity is switched and the capacitor will change the polarity of the plates as well.