Flashcarts became notorious with the introduction of commercial Gameboy mono and Gameboy Color flash cartridges (flash cartridges had been around for a while before but were normally only owned by those with a fair amount of electrical knowhow that had built them themselves or paid large price for them ($400 US was not unreasonable for a megadrive/genesis cart).
These were unpopular with the big companies such as Nintendo as essentially anyone with a PC and fairly minimal computing knowledge could obtain and play ROM images of games on original hardware (emulation had been around for a while by now and was equally being attacked).
Alongside this developers which hitherto had to pay quite large sums of money for arguably worse equipment than these flash cartridges were able to equip large amounts of their staff with these flash carts for development purposes which also caused revenue streams for Nintendo and such to fall.
To this end Nintendo took legal action and managed to secure a ban on imports (which was ultimately futile) of these flash cartridges into the USA as most of the carts were manufactured and distributed out of Hong Kong and similar areas. This being said a good deal of work did take place on flash cartridges within the USA.
This situation was short lived and by the time of the new GBA this ban was no longer applicable and flash carts once more began to come about. After a few very poor initial attempts at commercial GBA carts (they were not much more than re-engineered GBC flash carts which meant the ROM images needed speed patches as well as numerous save patches and a whole host of other problems plagued them) new carts based on good designs and equipment appeared thus really starting the GBA flash cart era. There exists numerous GBA flash carts and to detail them all would make for a fairly unreadable article, the pocketheaven wiki is a good resource if you wish to know about GBA carts.
With regard to the DS flash carts exploits were discovered (see this article for more on these exploits) that allowed the DS to run DS based code from the GBA port thus enabling people with minimal extra hardware to run DS code (homebrew and roms) with their GBA flash carts. This has since become the standard method although companies have started on the DS port side of things.
The term "the scene" as far as this article/wiki is concerned generally refers to the group(s) of people who partake in running of code be it homebrew (code written by people other than official developers) or copies of commercial code known as ROMS or ISO's normally via flash carts, it also deals the modification of consoles and also includes emulation of consoles.
Some consider it a façade for piracy and depending on who's definition you use (some people define "the scene" as the groups of people responsible for distribution of rom images) that could well be the case, most however define the scene as in the definition above.
It is generally centered around news websites which deal with things that happen that come under the collective interest of "the scene" as well as forum sites and IRC channels.
There is no offical language of "the scene" with numerous sites and channels having different standard languages:
A brief list and by absolutely no means definitive of general "scene" sites:
The sites above either provide news on roms releases and may also assign them numbers although these numbers can and do vary between the sites, they may also provide news on new emulators and code and tricks released as well as more general news like the shipping of new iterations of consoles. Many house/are forums where people can go and post questions and news of things.