A rainforest has a very technical definition, but colloquially it is very similar to a jungle. Jungles and rainforest have very dense foliage, plants, trees, and often distinct ecosystems that exist in different layers of the canopy/floor. Although some rainforests can technically not be in tropical latitudes, they generally are tropic, hot, and experience frequent rainfall. Rainforests are marked by very large amounts of rainfall.
Forests are usually in non-tropical areas and are more sparse, i.e. are less thick with regard to trees, foliage, leaves, and plants when compared with jungles/rainforests. You can often walk through a forest (with some difficulty), but jungles are extra thick and may require a machete to cut a path through the foliage.