The different types of Bump maps all have the same basic function of manipulating a model to make it seem as if it has some extra level of detail to it by changing how the light interacts with and bounces off the surface to make creases in the model that would be irrational to model on normally.
Bump maps tend to be grayscale images which are limited to 8-bits of color information. They basically work by defining the brightness of the image and using that to create bumps. The darker the colour of the bump map the further the indent appears on the object.
A normal map uses a RGB colour scheme that directly relates to X, Y and Z in 3D space. A tangent normal map is made up mostly of purples and blues and works best for models that need to deform in animation. For objects that don’t deform an Object Space normal map is better. This type uses many different colours unlike the tangent normal map.
Displacement maps physically display the object in the way that the map asks for. Like a bump map, a displacement map is also made in greyscale however it can be 16 bit or 32 bit granting it a greater range of colours to make more detailed maps.