There are numerous gods you can summon, here are 4 of them(there are six):
The Titan is the first obtained one, it is a giant creature with metallic rocks coming out of it. The Titan will be the first that you summon.
Second comes the Hyderean(aka Leviathan). It is an eel looking water creature. The Hyderean is the second you get to summon.
Third comes the Glacean (aka Gentiana). It is an ice fairy looking creature with the face of Gentiana, the woman who was responsible for Princess Lunafreya Nox Flueret in her childhood.
Fourth and last comes Ifrit, the fire god, It is a grumpy, fiery devil looking creature usually seen sitting in his throne. Ifrit is the last one you get during the story, the rest are obtained in chapter 15.

To summon a god, you need to HOLD the ALT button with a strong grip and NOT QUIT pressing it until the god is summoned.
Sadly you cannot choose the god, as the game summonds the god according to the situation it is in.
Lets take the Ifrit battle in chapter 13 for example, to defeat Ifrit you will need ice, therefore the Glacean must be summoned to defeat it.

I appreciate the time you took to read my guide! Have a nice day.