Slash-and-burn agriculture is a type of shifting cultivation where a segment of forest is cut down and the bits that aren't useful are set on fire. This has been a common practice in many parts of the world for millennia, but as humans become more populous and forests shrink, this practice is becoming unsustainable, and is no longer used in most developed parts of the world.
In Europe this was traditionally known as swidden agriculture, and this term is still used to refer to slash-and-burn in other parts of the world. In India and surrounding areas it may be called jhum or jhoom cultivation. The term milpa is sometimes used in South America, although milpa agriculture does not specifically require burning to clear land. The term slash-and-burn is particularly used to refer to clearing and burning the rainforest, in the hope that a clear statement of what is happening will help convince people to stop it.
Traditionally, a small, often nomadic family or community would cut a section of forest, allow it to dry for a few months, and then set it on fire. The ash would act as fertilizer until it was washed away by rains or depleted by a couple of growing seasons, and the group would move on to a new area. The land was not formally owned in the modern sense, so it did not matter if it took decades before it was recovered enough to cut and farm again. As long as there was significantly more woodland than human farmers, no irreparable damage was done to the forest.
While slash-and-burn methods are still common in many areas of the developing world, they are of particular concern in the rainforests primarily because there are so many other threats to the rainforest -- logging, the expansion of human settlements, cattle ranches, and large-scale farming all lead to large-scale clearing of the forest, to the point where even the comparatively small amount of damage done by small-scale slash-and-burn farmers is significant. As the forest area shrinks, more slash-and-burn farmers are forced into a smaller area and any given segment of forest is given less chance to recover before it must be farmed again. Farmers are also pushed into areas where the forest has already been damaged recently through logging or other factors, leading to a very poor soil quality, accelerated erosion, and decreased species diversity.
In addition to this, slash-and-burn may also be used to refer to new settlers coming into an area and clearing sections of forest for permanent settlements. In practice, this does not work well, as the soil of the forest floor may not be very fertile once cleared, and when cleared in large swatches these areas are likely to be very susceptible to erosion and desertification; while these are risks with the traditional slash-and-burn methods, the effects can be highly accelerated when done by newcomers who do not use the traditional methods, and particularly when they come en masse.
There are other negative effects to slash-and-burn agriculture. Erosion may release mercury from the soil, which then washes into local streams and rivers where it bioaccumulates in the larger fish. In some parts of the world, slash-and-burn agriculture is a common cause of wildfires. And the destruction and burning of forests tends to increase atmospheric carbon, which contributes to global warming.
When done responsibly, slash-and-burn agriculture can be good for local populations and for the environment. Swidden agriculture can reduce atmospheric carbon (PDF), provide a sustainable food source, and may, in theory, provide justification for indigenous peoples to maintain control over large areas of forest in the absence of active and constant use of the land. As it currently stands the legal framework for this is lacking in most parts of the world, but maintaining large areas of forest under the control of indigenous populations is one way in which we may be able to protect them from wholesale destruction.