Conditionals and branching in shaders

This is a quite wide subject which implies a lot of topics such as the platform or the compiler, but I think that it could be useful for everyone to know more about this.

A common mistake is to think that conditionals are bad when writing shader but the real bad thing is branching, right ?

I would love to know more about the difference between conditionals and branching with simple concrete examples and how to solve common issues.

Thanks