A trim router is a small tool used for flush trimming laminate and such. It does not require much torque because it cuts thin pieces. It is designed to be used with one hand and is easy to move around. It is not designed to handle standard routing chores and can quickly become unsafe (to user and tool) if too much is asked of it.
A fixed router can do all the above and standard routing chores. However, a fixed router must set the depth prior to cut and this somewhat limits its use, such as in routing away wood under a CP for the joystick (this is a job for a plunge router). Fixed routers do edges only.
A plunge router can do all of the above, and start a cut in the middle of a work surface. You set a stop for the pre-determined depth of cut, align your router over the work surface, then plunge down to the stop and begin your cut. In addition, a plunge router can function just like a fixed router for edge cutting tasks.
All in all a plunge router is best if you can only own one. Then either of the two above are good second choices. Many times you can safely use a smaller router for bigger tasks if you take multiple small cuts each pass to attain the finished cut. I have successfully bought many tools from Harbor freight for limited use. They will do the job, but for how long is uncertain. Good luck with your tool buying.