To summon your pet double click it, in your inventory. To unsummon your pet, right click and choose "Unsummon Pet" or "double" click your pet icon in the inventory.
Please note that when you unsummon your pet it loses 1 minute worth of exp.
Each second your pet is out, it looses HP
Rank HP Death w/o Feed
D 180 3 mins
C 300 5 mins
B 420 7 mins
A 600 10 mins
S 1500 25 mins
Your pet gains EXP over time. NOTE that just cause the S pet no longer gains exp it doesn't mean that it won't die of hunger. It will still lose hp while you have it out, so you still need to feed it to keep it alive. However as mentioned in the part above this the S grade pet has like 25 minutes of life, so you don't have to be very stressed about it. Below is the EXP that is earned by class level.
Time Rank EXP Total Feed Total Time
1 min D 0.166% 18,072 10.04 Hours
1 min C 0.083% 36,144 20.08 Hours
1 min B 0.041% 72,288 40.65 Hours
1 min A 0.02% 144,576 88.33 Hours
Note that when you unsummon your pet, it will lose approximately 1 minute worth of exp. This is better than how it started because at the launch of v9, pets would lose 10 minutes of exp, which was very annoying. Additionally before the maintenance after v9 where they lowered it to 1 minute, S grade pets used to lose some of their HP (instead of EXP) when you unsummon them however they were changed to lose absolutely nothing.
Time Rank EXP
1 min D 0.16%
1 min C 0.08%
1 min B 0.04%
1 min A 0.02%