Your party is your character plus two creatures. Creatures live in the soul bag; the two you seat fight at your side and share the gear screen with you.
1Your two creatures
The lower half of MENU > EQUIPMENT is your party (1): each creature with its level, stats, moves and its own gear rows.
2Seat a creature
Click a creature's panel: the one seated now is under CURRENTLY SLOTTED (1) and every other creature in your soul bag is under AVAILABLE (2) - click one to seat it in that slot instead. A party slot is never empty: a creature can only be replaced, not removed.
3Souls on gear
Every gear row ends in a soul slot (1): a creature from the soul bag seated there lends its power - a move or a passive - to the character. The soul slots open at floor 12.
4The class soul
Your class soul (1) is handed out at creation; click it to see what it does and to upgrade it once the souls rung is open.
Screenshots are taken from the live game by tools/tutorial_shots.py (dfmc-client) and regenerated after interface changes. The rules behind each screen: system-party, system-equipment.