If we consider PLC types to be based on their size and features,
they can be placed into classes:
Pico or Nano, Micro, Mini, Standard, RTU, Safety, OEM, and PAC.
Keep in mind though, trying to classify PLCs into types is not easy as there is an overlap of features among them.
Some may suggest the DCS (Distributed Control System) is a type of PLC. Originally the two where more distinct, the DCS for process/batch and PID loops and PLCs generally used for discrete or on and off control. However the PLC has evolved over the years to become very powerful controllers available to perform the many of the tasks that the DCS can do. For now, I choose to exclude this one from classification in a PLC type, but you can certainly argue the case.
It is difficult though identifying the differences of PLCs in the Pico or Nano class and the Micro and Mini. A PLC manufacturer may name a particular PLC line Pico ( for example Allen Bradley), while another competitor will name their line of similarly featured and sized model as Nano (for instance GE).
Among PLC types is the RTU or Remote Terminal Unit which are specialized for communicating remote measurement and process data to a SCADA systems usually. Since these devices may be placed in the middle of nowhere they are design to consume little power (possibly solar power) and endure environmental extremes. This one might be a bit of a stretch if is not actually controlling anything, but just relaying measurement data to a SCADA.
Safety PLCs are designed with some form of logic processing redundancy and monitoring as well as input and output self checking. As you may have guessed, these cost more, roughly 30% higher than the standard fare and are use in high risk situations. These controllers are typically used for Safety Integrity Levels (SILs) 2 and 3.
OEM PLCs are generally without a case or enclosure and therefore suited to fit inside a product that is mass produced, for instance a washing machine.
PAC or programmable automation controllers are essentially the tops of the PLC food chain. They are very powerful in terms of processing speed, extensibility, programming and communications.
Again I'll say that the RTU, PLC, PAC and even DCS lines are blurring as these once specialized devices adapt more of the features as the others. This is simply due to the never ending advancement of technology and the once expensive cutting edge technology becoming more accessible.