A software interrupt occurs when the processor executes an INT instruction. Written in the program, typically used to invoke a system service.
A processor interrupt is caused by an electrical signal on a processor pin. Typically used by devices to tell a driver that they require attention. The clock tick interrupt is very common, it wakes up the processor from a halt state and allows the scheduler to pick other work to perform.
A processor fault like access violation is triggered by the processor itself when it encounters a condition that prevents it from executing code. Typically when it tries to read or write from unmapped memory or encounters an invalid instruction.