You get to have a summon when your level 20 (use to be level 30 in Global Goonzu). Keep in mind that you do not need any capture skill to have a summon or a summon skill. Once you reach level 20 and would like to have a summon, go to a market or to a capture artisan to get a Evil Raccoon summon doll. Double click the doll and there is your new summon. Keep in mind that the doll sometimes fail so make sure you have a few ready in case the first one doesn't work.
Every time you summon your Evil Raccoon, it gets a portion of the experience you get. So to level it, make sure it is alive and summoned. The summon also has a limit of time it can be summoned. If this time runs out, you will need to wait a certain amount of time for the recovery period before you can summon it again.
Horses and other mounts are summoned from medals. The Vixen is summoned from a spirit orb. Starting with the March 5th patch, users who reach level 15 (after the patch) receive a brown foal. Its stats are already maxed out and it cannot be traded. But you can receive dormancy points which can give you a various amount of items. You can earn dormancy points by being online for a extended period of time.