The short and perhaps the most efficient non-technical answer is that CrossFire is a high-performance PC gaming graphics platform made possible by the use of multiple graphics cards coupled with a CrossFire enabled motherboard with a single CPU, in order to increase graphics performance and quality. So, in layman terms you can essentially run applications like high-end games at a much higher frame-rate than it is possible with a single graphics card.
Previously called CrossFire, AMD CrossFireX (as it is called today), refers to a brand name for the multi-GPU system by Advanced Micro Devices which was in fact originally developed by ATI.
Crossfire-X technology allows you to connect two or more of the same family GPUs together for combined performance. The great thing about CrossfireX is that you do not slow down the faster GPU's clock rate when running it in conjunction with other graphics processors. So for example, if you run a Radeon 7950 and a 7870 together in CrossfireX configuration, that is fine. This differs from Crossfire and SLI which required you to pair the same GPUs together.
Since it was the first of the two technologies to be released. SLI was originally introduced by 3dfx in 1998 with their Voodoo 2 card. At that time SLI meant Scan Line Interleaving and worked by making each GPU to process one group of lines (one GPU processing odd lines and the other processing even lines). NVIDIA bought 3dfx on April 19th 2001 and introduced a similar but updated concept for their video cards in June 2004, renaming SLI to Scalable Link Interface.
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