Here's my understanding. Bear in mind I didn't grow up in the US system so there's quite probably errors. Even so I think I have the general idea.
Broadly, most civil aircraft are certified (jeez, I loathe the word 'certificated' which seems to have become the norm. Dunno why, but I do) under one of two US sets of regulations - Federal Aviation Regulation 23 (FAR 23) or Federal Aviation Regulation 25 (FAR 25). These, in turn are a subset of Part 14 of the USA Code of Federal Regulations ie the overarching set of US Fed. Regs. from which all the others gain their authority.
FAR25 deals with Transport Category. This set of rules specifies with great detail what requirements must be met for an aircraft to be certified to comply with these rules. A general rule of thumb is that FAR25 aircraft must be able to have sufficient performance, and constrain their operations at all stages of flight, to either reject the take off or continue flight to land at a safe airport. Additionally, there are requirements about redundancy to systems & procedures to prevent any forseeable single point failure leading to a significant reduction in safety.
Pretty much any airliner is a FAR25 aircraft.
FAR 23 specifies the requirements for an aircraft to be certified under its set of specifications. Generally aircraft certified under this part do not have to meet the same level of performance or redundancy that FAR25 aircraft must meet. Most general aviation aircraft from pistons to turboprops fall under this Part.
However, within this Part are subsections, one of which refers to commuter category aircraft. This subsection tries to give some semblence of FAR25 aircraft performance & redundancy, but without having to go through the entire FAR25 rigamarole. It's intended to cover aircraft that were never designed to be FAR 25 type aircraft but are used similarly ie Regular Public Transport.
This isn't meant to imply that getting an aircraft design certified under FAR 23 is simple or easy. It's not, but in general meeting FAR 23 requirements is less onerous. For example, a multi-engine aircraft has no requirement to have asymmetric climb performance except in an aerodymically clean configuration at maximum continuous power.
Special FAR 41 (no longer current) was a special FAR (hmmm....what gave that away?) that allowed certain FAR 23 types to gain a supplemental type certificate that went a little way towards FAR 25 safety. It's no longer current and, I believe, has become the FAR23 (Commuter) requirements.