Idols are Statues of gods located in The Hub. These Idols may be prayed to and offered Dream Spheres in exchange for stat increases and other benefits.
The Steam store page describes meta-progression between runs; the idols are where Dream Spheres are spent on it.
A strategy guide recommends spending on damage and health first, then the summon-limit statue or the wizard statue if you cast often, and says the statue nearest the bed is the one that repairs your sword.
You pray to your gods by speaking to their idols in the Mind Prison, offering Dream Spheres found in the dungeon.
The manual states plainly that these upgrades are permanent for your save file and persist between runs.
From the game's official manual. Unlike the rest of the notes on this page, this is a primary source: it is the developer's own documentation, not community research.
