You can purchase many items from the Darkmoon Faire with this, see the currency page for the full list.
Tickets are earned by doing various quests or playing the games, see the currency page for the full list.
These tickets are rewarded for doing quests at the Darkmoon Faire. Some quests can be completed once per Faire/Month, while some are dailies and can be done every day. The full list of quests that rewards these tickets is here.
They can be used to purchase a number of things: pets, mounts, heirlooms and miscellaneous fun items. Please see the full list here.
You can earn up to 145 of these per Faire, assuming you do all the monthly quests (including turn-ins) and the dailies every day the Faire is active.