I presume you're talking about a "programmer" as a piece of equipment, rather than a person who writes programs...? ;-)

A programmer is a piece of equipment that loads "programs" into chips. The chips could be programmable memory devices (PROM, EPROM, Flash, etc), programmable logic devices (PAL, PLD, CPLD, FPGA, etc), Micrcontrollers with on-board programmable memory - in fact, anything that needs programming.

An emulator - or In-Circuit Emulator (ICE), to give its full name - is a much more sophisticated piece of equipment that replaces a processor in the target hardware and "emulates" its behaviour in the circuit.
As well as emulating the processor, it also gives you access to all the internal working of the processor (registers, memory, etc), can record trace information, set breakpoint, etc, etc - all without consuming any program resources!
As Erik says, an emulator may also have the facility to program the real chip.

These days, some modern processors (including some 8051 derivatives) have a certain amount of debugging hardware built into the chip. This is often accessed via a JTAG interface to give you almost the equivalent of an ICE, without all the expense and the hassle of having to remove the real processor and fit the ICE "pod".