When a hardware interrupt occurs the CPU stops executing the instructions that it was executing and jumps to a location in memory that either contains the interrupt handling code or an instruction branching to the interrupt handling code. This code usually operates in a special mode for the CPU, interrupt mode, and, normally, no other interrupts can happen in this mode. There are exceptions though; some CPUs rank the interrupts in priority and higher level interrupts may happen. This means that the first level interrupt handling code must be very carefully written and it often has its own stack, which it uses to store the CPU's execution state (all of the CPU's normal registers and context) before it goes off and handles the interrupt. Some CPUs have a special set of registers that only exist in interrupt mode, and the interrupt code can use these registers to do most of the context saving it needs to do.
When the interrupt has been handled, the CPU's state is restored and the interrupt is dismissed. The CPU will then continue to doing whatever it was doing before being interrupted. It is important that the interrupt processing code is as efficient as possible and that the operating system does not block interrupts too often or for too long.
