The AP shop is an area where you can buy special items through the use of AP (Real Money). You can buy AP by going to the main alaplaya site and clicking on account, it should take you to the appropriate page and you need only select a payment method and amount to purchase from there. Please note that with higher purchases you may get something called BP as well, these have no use in the game itself.
The items offered consist of Entry Change Tickets, which you use to change out your character in battle. Skill Scrolls, which are used to unlock skills for a character. Costumes, which change the look of your character and give you an increase to the amount of coins you recieve at the end of a game (as a special note these do stack as long as the character wearing the costume is in your set of three you get the bonus from them, this can be a handy, if expensive, way to get a good amount of gold quickly.)
You can also purchase mission boards of varying sizes, these mission boards give out everything from tokens to coins but are somewhat time consuming in the cases of the longer ones.
You can also choose to change your nickname if you wish but it is very pricey to do so, meaning you may wish to pass on it for now unless you really feel the need for it. In the future it appears that there will be items that can reset many of your other stats but it remains unconfirmed for now.
You can also purchase extra slots for your common inventory on the hero managment page by selecting the expand button on the top right corner of your inventory.
Finally and most importantly in the section of the shop titled Hero Card you can purchase permanent and trial hero cards. The trial hero cards can be purchased with normal in game coins but the other cards require you to purchase AP in order to get them. You can also purchase the newer heroes here for AP, though they cost more than the normal card.
As a special note anything bought here can be sold and bought on the market once you get to the level of vanguard. You can purchase beforehand but it won't be worth it for the most part.