The temperate zone is that part of the earth that lies between the arctic zones (north and south) and the tropics. The northern temperate zone spans the area between the Tropic of Cancer at about 23.5 degrees north and the Arctic Circle at 66.5 N. The southern temperate zone is between the Tropic of Capricorn at approximately 23.5 S and the Antarctic Circle, 66.5 S.
The word 'temperate' means 'between two extremes' or 'moderate', and the temperate zones are indeed zones where the climate is between the two extremes of arctic cold and tropical heat. The temperate zones are characterized by having four seasons and roughly equal-length summers and winters.
A deciduous forest is one comprised primary of deciduous trees - those that lose their leaves once a year. A temperate deciduous forest is one that is in a temperate zone. The 'temperate' modifier is redundant, although often used, because there are no deciduous forests in the tropics or the arctic areas.
If you grew up in America or Europe on stories of Robin Hood and Daniel Boone, you probably have the idea that deciduous forests cover more of the planet than they actually do. Not only are deciduous forests only found in the temperate zone, they're only in the northern temperate zone. Their total area on the earth is considerably less than the total area of grassland.
Deciduous forests are homes to a vast array of wildlife. All the hibernating animals live in temperate zones; in the tropics, there is no need to hibernate, and in the arctic, the summer is too brief for an animal to do all the 'living' it needs to do to reproduce and prepare itself for a lengthy sleep. Because of the suitable climate and the desirability of deciduous trees for building, the areas occupied by temperate deciduous forest were widely settled by mankind, with the result that the forests themselves are being lost to farmland.
Fortunately, many of the remaining forests in the US and Europe are now national parks and nature preserves, which ensures that future generations will be able to enjoy them.