When I was small, British Jews greeted each other on festivals with "Good Yomtov." It was one of those generic, Jewish expressions which could speed your passage through the El Al security check.
Now they mostly use the Hebrew "Chag Sameach" instead.
Sameach means "happy" or "joyful:" chag is a "holiday." The Talmud (Chagigah 10b) argues about the original meaning of chag when it is first used in the Torah (Exodus 12:14). One view is that it means a "sacrifice." This interpretation was preserved through the chagigah offering in the Temple every Pesach, Shavuot and Succot.
The other reading is that chag signifies eating, drinking and celebrating. This is the way that Jews observe the simchah of chagim now that we have no Temple; by putting on our best clothes, and if possible buying new ones, and by serving the finest food and drink that we can afford.
An essential aspect of the rejoicing is that we invite the poor and lonely to celebrate with us. As Maimonides scathingly puts it: "If someone locks his doors, eats and drinks with his wife and children and doesn't feed the poor and bitter of soul, then it isn't the simchah of a mitzvah; it's the simchah of his stomach" (Laws of Yomtov, 6:18).
Even though it's a mitzvah to be sameach on every chag, there is even more simchah on Succot than on the other chagim. The joy of gathering the harvest together with the relief of having our slate wiped clean on Yom Kippur combine to create the greatest simchah on this chag. Chag Sameach!