Emulators are applications which reproduce another, different computer system in software allowing the software from one machine to be run on another. Emulators are used for a wide range of purposes. For example, OS X provides emulation on x86 computers to allow old PowerPC binaries to be run. Microsoft provide an ARM PDA emulator for testing applications without having to repeatedly copy things onto a device.
One of the most popular uses for emulators however is to play video games from older (or in some exceptional circumstances, current) systems on modern hardware (generally PCs but the Pandora, GP2X, GP32 and PSP are also popular platforms for running emulators). Nintendo sells a number of official emulators through its Virtual Console service for the Wii.
Pandora and emulators
One of the popular uses for the Pandora is to run emulators of old video games systems (though some old home computers and calculators are also emulated). The Pandora is suited to this task because it has a fairly standard control layout similar (or almost identical to) the layouts of the input devices of the systems it emulates. Furthermore, it has an LCD of resolution 800x480, which is large enough to accommodate most older systems without any downscaling. Of course, it is also portable meaning you can play these games whenever and wherever you want without having to carry a TV, machine and collection of cartridges.
What is needed to run emulators
In most cases, emulators can simply load software and run however some require a dump of the BIOS ROM stored within the system itself. These dumps cannot be distributed legally without the permission of the company/individual who created them due to them being copyrighted. In some cases (such as the GBA) it may be feasible to extract a dump of the BIOS yourself if you own a GBA and the necessary equipment but for others (like the PlayStation) the only way most people can optain a BIOS dump is to illegally download it. Each system has a unique BIOS, though many older ones don't have a BIOS at all, or it is emulated along with the hardware. You should check the individual pages for each emulator to see if they need a BIOS dump to run.
An emulator is no use without software to run on it. These normally come in the form of ROM dumps (since they are extracted from the ROM chips that used to be used to distribute software) generally known as just "ROMs". A few ROMs are "homebrew", that is, free software or freeware. However, ROMs of commercial games are illegal under copyright law unless you dump them yourself (and even then you are only allowed to make use of them under very specific circumstances). ROMs for one system will not work on another system's emulator, just as you can't plug Super Nintendo cartridges into a PlayStation and run them. As such, you need ROMs for each system and an emulator for each system (though some special cases exist such as MAME which will run ROMs from multiple arcade systems).
What the Pandora can emulate
The Pandora, despite its apparent low clock speed when compared to those of modern desktop PCs, is a powerful system (clock speeds cannot be reliably compared between systems). Although there is very little hardware for accelerating graphics functions, the CPU is fast enough to emulate many systems purely in software.
Most video game machines up to the 16-bit era (i.e. SNES, Mega Drive) could in theory be emulated on the Pandora, and some 32-bit consoles have the potential to be emulated with varying degrees of success. That does not mean that everything before this point will be emulated as there are hundreds of machines, some of which are virtually unknown and others which have unusual hardware which is difficult or impossible to emulate.
This means that there are literally tens of thousands, possibly hundreds of thousands of games which can be played on the Pandora if you are able to obtain the software.
Where to download emulators
Most software (including emulators) for the Pandora can be obtained from the Pandora File Archive. You will find the emulator section here.
This wiki houses compatibility and performance lists for various emulators available on the Pandora. You can find these on the Emulator Compatibility page.
Please do add to these lists as you test more games so that others may benefit from settings you may have discovered (for some users a particular game running well is enough to justify the purchase).
Here is a list of systems for which emulators are available on the Pandora (it may not be complete as new software is being released regularly). This doesn't take into account how complete the emulators are, merely that they exist. For performance data, look at the compatibility database.
For a list of announced projects, please see Prophet's list of potential emulators and development status on the GP32X forums.