Crash shifting (also sometimes referred to as blind shifters, 116-style shifters, sliding gear shifters, and gear munchers) is when you have gears slide to engage a different set of gears like so:
Dog shifting (also sometimes referred to as 45-style or 716-style shifters, depending on the dog design) engages a set of gears by using a 'dog' to lock to the shaft one of a pair of freewheeling gears. Like so:
The orange part is the dog, green and blue are the freewheeling gears, red part is the output shaft, and the yellow part is the shifting rod that actuates the sliding action of the dog. (The shifter appears to be in the middle of shifting and the dog isn't really engaging either of the gears)