A COMP is a portable computer used to summon Demons with the Demon Summoning program. While the term Demon has a certain connotation, in this case it means any kind of supernatural creature (including those found in whatever setting you go to), meaning that things like angels and spirits are also considered Demons by the COMP. Demons that you have a contract with can be fused to create more powerful Demons, giving you an incentive to make contracts with as many Demons as possible. Thankfully, they mostly seem to ask for money and healing items, but if you have good charisma and know how to "read" a Demon's personality, you can often bind a Demon for free.
It also comes with an automap & GPS, any features found on a real world cell phone, magically encrypted email, and various trackers for your quests, items, companions, and abilities, a scanner and database for things you've summoned or encountered, and the ability to purchase apps that can upgrade the COMP or add new abilities.
You can choose any COMP from the series, including weird and wacky ones like the GUMP.