Programmer is an ambiguous term. If it refers to a living person it could be a person who writes programs for computers, a worker, or a person who uses psycological means to produce desired responses, a manager. If it refers to a piece of electronic hardware, then it is a device that transfers a stream of bits from one place to another. Usually the source of the transfer is a PC or other computer on which the worker developed his program under the coercion of the manager, and the destination is an embedded processor that will subsequently perform some desired function.
An emulator is a person who tries to be something they are not, or it could be a piece of hardware that looks like a processor, has memory like a processor, and executes instructions like a processor, but is not a processor. It often has the advantage that you can probe points of the circuit that are not accessable inside a chip. Generaly it is a combination of hardware and software. They are often quite expensive.
A debugger is a person or program that finds and fixes bugs in programs. As a program it usually takes the form of a small monitor that runs on the target processor, and can examine memory, install breakpoints, and trigger on various conditions. Sometimes it can allow changes in programs for fixing a problem and trying again.
The lowest cost method of program development is called Edit, Assemble(Compile), and Burn. This expression comes from the days when programs were developed with progams called editors, assembled with programs called assemblers, and programmed into PROMS with devices called PROM Burners. They were called PROM Burners because they used a current pulse to burn out and destroy a polysilicon or metalic fuse for each zero bit.