Imagine Puzzle & Dragons. You pick a dungeon to go into, you pick five monsters plus a "friend" to form your party and then you go into the dungeon.
Now, you move your monsters around in the dungeon by literally flicking the rooms in the direction you want to move (moving left means you flick left to right, moving up means flicking up to down, and etc). Battles are played in the classic DQ fashion.
At the end of a battle, you may recruit a monster. The point of each dungeon is to defeat the boss, which you may or may not recruit. You can also get "maps" to recruit monsters and you get these maps by using the vending machine. Sometimes, Square-Enix will give you a gift for whatever reason (100 million downloads, whatever).
There are events to play as well which can give out special prizes. These come and go at whatever time Square-Enix chooses.
It's pretty fun as far as F2P goes. There are timers and you can't really play indefinitely in one sitting due to a stamina system.
There's a monster fusion system that involves the egg monsters, except they'll have different patterns. Collecting these and fusing them in a specific combination can yield a new monster as far as I understand it. You can also sacrifice monsters to strengthen one.