Triangle congruence criteria are rules that determine when two triangles are congruent, meaning they have the same shape and size, with corresponding sides and angles being equal. The primary criteria include Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Right Angle-Hypotenuse-Side (RHS) for right triangles.