“Rare modules” are the blue second-tier modules. Early game these may be utilised for your tower, but eventually these are largely going to be used as fodder for Epic Modules and for auto-shattering to acquire Shards (10 shards per rare).
When purchasing modules for gems, rare modules have a 29% chance of appearing.
Bosses can drop rare modules (starting at 0.5%), which is increased with the Rare Drop Chance lab.
Rare modules can be merged up to Legendary Modules.
The starting rarity determines how far a module can be merged and what rarity that module can be merged to.
PS! Due to the fact that Rare Modules can be merged up to Epic, starting out with the game it can be confusing or hard to differentiate these compared to the modules that have started as Epic. One visual cue is the central part the module graphic itself (not referring to the frame). I.e., a module that started out as Rare will have a graphic in the center that is blue and yellow. Whereas a module that started out as Epic will have a graphic in the center that is usually pink or red.
An example of a module starting as rare, merged to Legendary+.
An example of a module starting as rare, merged to Epic.
An example of a module starting as epic.