Forests generally describe an area with many tall trees but with clear, traversable land. Jungles generally describe an area with tall trees but with thick vegetation, that is difficult is not impossible to traverse. A forest may be dry or rainy, whereas jungles are generally humid or tropical.
A jungle is essentially a dense tropical forest and has a more diverse ecosystem (a greater variety of plants and animals). The Amazon rain forest is a perfect example.
Woods vs Forest
Woods generally describe an area similar to a forest, but smaller. There is no clear delineation as to when a dense growth of trees is too large to be considered woods and is therefore a forest.