LindeX is a service created and run by Linden Lab that allows residents to trade L$ (Linden Dollars) for USD$ (United States currency) and vice versa. LindeX is not used to buy currency like other services such as IGE. The money traded on LindeX is closely watched by Linden Lab and used in creating economic statistics as well as judging if the value of the L$ is inflated or deflated.
Its conception was heavily surrounded in drama as LindeX is very similar to a resident created service called GOM (Gaming Open Market). GOM was a currency trading website that would show live trends and statistics based on the trading market at that time. Originaly intended to be for multiple MMORPG and virtual worlds, GOM ended up being restrictively SL based, this was due to other companies not giving support for fraud situations. Linden Lab was alternatively very supportive of GOM, which was also part of GOM's decision to be strictly SL based. GOM quickly caught on with residents and became the most used service for currency trading in SL. GOM's trading statistics were also used by many people to gauge how well the economy was doing. Months later Linden Lab for some reason decided to offer its own currency trading service which was very similar to GOM. They named the service LindeX as in "Linden Exchange" (not windex). Anticipating that residents would most likely go to LindeX (an official Linden sanctioned and run service) and that LindeX would be built in to SL, GOM closed shop thinking they would be no match for the service. Many users were angered by LindeX, reasoning that LL broke an unspoken rule of not competing with its users as well as directly ripping off the GOM service.
Note that when buying and selling Linden Dollars (L$) through LindeX, residents are not trading directly with Linden Lab; rather, they are trading with OTHER residents and can set buy/sell prices and limits at their own discretion.
Regardless Linden Lab pursued the project and LindeX is now widely used. LindeX can be accessed via the website or through the Second Life client (by clicking your L$ value at the top of your screen or clicking the L$ button next to it).