Bump maps are grey scale images the only have 8-bits of information which means it’s only 256 different colors of black, grey, or white. These values say whether it’s up or down. The whiter it is, the more it comes out of the surface, and the darker it is the further it goes in. Bump maps are easy to create in Photoshop but break easily if viewed from the wrong angle.
Normal maps are a better version of a bump map, there is also no addition resolution added to the geometry but they work differently. A bump map used grey scale values to provide information. A normal map uses RGB that works with X, Y, and Z axis in 3D space. The RGB information tells the exact direction of the surface normals. The most commonly used map is a Tangent space map is a mixture of purple and blues. These work best for animation such as characters or assets that won’t deform. The other one is an Object space normal map, which have a rainbow assortment of colors which are slightly better than Tangent space maps. Normal maps are more difficult to generate in Photoshop, normally you will bake a normal map out of a high poly mesh.
For more detail on low-resolution meshes, displacement maps are the best for this. These maps physically displace the geometry, these are rendered real-time. They can be baked from high-resolution model or painted like a bump map. You can use an 8-bit map, it’ll be better if you used a 16-bit, or 32-bit map. Creating the geometry in real time can be exhausting on the system, so they just calculate this at render time.
You can use these maps together if you wanted to, there can be added benefit, but you should just use the map that best fits your needs.