What's the difference between a FP, gain guided and DFB laser diode?
Fabry-Perot laser diodes are lasers whose mirrors are simply the flat cleaved surfaces at the ends of the laser chip.

Distributed feedback (DFB) laser diodes are lasers that have a grating structure in the cavity that produces multiple reflections throughout the cavity. This leads to narrower linewidths than are produced by FP lasers.



(Image source: Laser Focus World)

Another type you didn't ask about is the Distributed Bragg reflector (DBR) laser. A DBR typically has two separate grating regions, on either side of the gain region. They're usually distinguished from the DFB because the grating doesn't overlap the gain region, although there is some grey area and the terms are not always used consistently.



(Image source: US Patent #6638773)

Where FP, DFB, and DBR types describe how the longitudinal reflections that provide laser feedback are produced, the designation gain-guided describes how the mode is confined in the transverse dimensions. In a gain-guided laser there is no waveguide patterned between the reflective structures (end facets or DBR regions). Instead, we rely on a narrow region where current is injected to maintain the transverse mode. 

This works mainly because any transvese modes that don't overlap the injection region will see much lower gain and so won't lase.