From our perspective as people walking through or standing in a forest, the level of light may vary. When a forest is dark, this is caused by something called the canopy. In forests, the uppermost layers of branches and leaves may be so densely packed that it prevents much light from shining down to the ground. Where there is especially dense growth, as in a rain forest, things can be quite dark indeed! The canopy may offer shade to animals who live below, as well as protecting some smaller plants from too much sunlight. That being said, the canopy may also "starve-out" plants which are competing for sunlight.
Of course, if it is nighttime and the sun is not shining, a forest will be dark regardless of how dense tree growth may be.