When cut down after mining, resets yield to whatever remaining terrain provides (either that of a Hill or whatever flat land is beneath it), and provides a lump sum of Production to the nearest city of the Worker's civilization. Provides +1 bonus Faith if unimproved and next to a Celtic city (+2 bonus Faith with 3 or more unimproved tiles).
Forests present interesting opportunities, such as some unique resources like Deer and Truffles, and a natural defensive bonus. They also provide a one-time boost to a nearby city's Production process when cut down, which can be used strategically. Lumber Mills, Camps, and Trading Posts can be constructed in forests without cutting them down, but ordering any other tile improvements will cause them to be removed (still providing the one-time production boost).
Forests on certain terrains are very important, because they replace the base yield and so in certain cases are effectively "improvements" to the base terrain yield. Try to preserve them there should you settle a city in the tundras or on grassland. They may prove a hindrance or help on hills, depending on the yield required – chopping them down and mining them may prove helpful.
Also, the Celtic and Iroquois civilizations have unique bonuses and buildings which make a very effective use of Forests - when playing with them, search out and settle in or near forests. Note that the Iroquois do best with an advanced start where the lumber mill improvement is already available and may suffer in earlier Eras without it.
For early man, the forests were a great source of bounty, providing wood for fire, tools and shelter, and also many animals for sustenance and clothing. As a city grows there is a great temptation to cut down the forests for farmland, but a wise leader will always leave some standing - for productivity and to lift his people's spirits. Also, military units stationed in forests receive a significant defensive bonus.