Emulation is hardly a new idea. In the days of the old 8-bit machines, there were all kinds of books, articles, etc. on how to convert a program written in one form of BASIC so that it could run on another. (Example: converting an Applesoft program to run on a TRS-80). There were even programs that would try to modify a BASIC program to run on another machine.
And of course there were the hardware solutions; CP/M cards that fit into Apples or TRS-80's, Apple cards that went into IBM PC's or Macintoshes, various hardware emulators that would make an Atari ST emulate a Mac, or make an Amiga emulate a Mac, or make an Amiga emulate just about anything else....
If I recall correctly, the Amiga was the first consumer computer really fast enough to make software emulation somewhat practical. Remember the Transformer? Of course it wasn't all that fast (about 1/4 as fast as a "real IBM"). A bit later, Amiga developed the "Sidecar", which *was* a "real PC" controlled by the Amiga. With the Amiga 2000 came the Bridgeboard, and similar third-party solutions.
On the software side, I recall a Commodore-64 emulator for the Amiga (Anyone remember the proposed "Emily-64" emulator that would simulate an Apple, an Atari, or a C-64 on the Amiga?). There was also an Apple II emulator for the IBM PC (286 or better). These came out the early 1990's, but never really caught on.
(A big problem with getting any of these software emulators to work was access to the emulated computer's internal software (ROM). Most of these were (and still are) copyrighted and illegal to distribute. So, the emulator programmers gave instructions for downloading the ROM contents of the original machine onto the machine running the emulation. Since disk formats were generally not interchangeable between different makes of machine (e.g. an IBM PC couldn't read an Apple disk), a special serial port connection (null modem), and additional software were often required.
Emulation took off in a big way with the advent of faster computers such as the Pentium and Power PC. These fast processors made it possible to emulate older 8-bit machines (Apple II, Commodore 64, etc.) with the same speed and features as the original computers. Also, early-80's video games could be emulated in the same way, as those games used the same types of processors (6502, Z80, etc).
As Pentium and Power PC speeds increased, emulation of more complex machines (Amiga, Atari ST, Mac) and newer video games (Nintendo, etc) became practical.