Two personas, different arcana
The fusion chart gives the result's arcana. Add the two base levels, divide by two, round down and add one. You get the lowest persona of that arcana whose base level is at least that number. If no persona is that high, you get the arcana's strongest.
Example: Arsene (Fool, level 1) + Pixie (Lovers, level 2) gives a level target of 2 in the Chariot arcana, so the result is Agathion (level 3).
Two personas, same arcana
Two personas of the same arcana always give that arcana. The game takes the averaged level plus one and gives you the strongest persona at or below it, skipping both ingredients.
Treasure personas
Royal has 9 treasure personas, rare shadows that appear in Palaces and Mementos. You can't fuse a treasure, but you can fuse with one. The result keeps the other persona's arcana and moves one or two places up or down that arcana's list, depending on the treasure. For example, Jack Frost (Magician, level 11) + Regent gives Nekomata (Magician, level 17). See the treasure chart.
Special fusions
24 personas only come from special fusions with a fixed list of two to six ingredients. See the special fusion list.
Locked personas
The strongest persona of each arcana needs its Confidant at rank 10. A few more unlock through story progress, and some are DLC. The calculator's recipe filter can hide all of these.
Your level matters
You can't create a persona whose base level is above your own level. If a recipe gives a persona you can't make yet, level up first or pick a lower recipe.