We definitely have a case of blossom-end rot...

Symptoms:
The bottom of tomatoes that are beginning to ripen turn black or brown.
Blossom-end rot begins when the fruit is about half grown, and continues to
develop as the fruit matures. The first symptom is a water-soaked area at
the blossom end (opposite the stem end) of the fruit. This spot will grow
larger, turning dark and sunken. It is usually worse on the first fruit
cluster but can be a problem throughout the season.

Pictures:
http://www.ifas.ufl.edu/~ftsgweb/BERDBC.htm
http://www.oznet.ksu.edu/dp_hfrr/extensn/problems/blendrot.htm
http://130.111.117.57/images/diseases/tomato.jpg

Edibility:
Tomatoes that are already affected will continue to be deformed. They are
safe to eat, but the disorder reduces the amount of tomatoes that can be
used. If you have a lot of tomato plants, it might be best to pick off and
destroy the fruits that are severely affected.

Cause:
Water-related:
Blossom-end rot is not caused by a pathogen, so it is not an infectious
problem. It results from a calcium deficiency in the plant caused by large
fluctuations in soil moisture. When soil moisture is limited, plant growth
slows and nutrient uptake by the roots is reduced. If water becomes
available again, from rain or irrigation, the plant begins to grow rapidly
but the uptake of calcium lags behind. In this way the rapidly expanding
fruit tip does not have enough calcium available to develop properly, even
though there is plenty of calcium in the soil.

Soil-related:
Another cause of blossom end rot is over-fertilization, especially of
nitrogen, which stimulates vegetative growth. Excessive vegetative growth
increases the transpiration surface and further prevents calcium
accumulation in the fruit. Tomato varieties with large amounts of foliage
tend to be more susceptible to blossom end rot because the leaves require
calcium. Adjust the nitrogen rate for each cultivar to reduce blossom end
rot. Avoid ammoniacal forms of nitrogen that compete with calcium during
uptake from the soil.

Treatment:
Water-related:
The best method for controlling blossom-end rot is to maintain even and
adequate levels of soil moisture. In such conditions, plants grow at an even
rate and the nutrients stay in balance. Do not let the soil dry excessively
between waterings, as this stimulates blossom-end rot. Mulch to conserve
moisture. Mulching plants with straw, newspapers, or thin layers of grass
clippings can help to control blossom end rot.

Soil-related:
Check soil pH and soil nutrient levels annually and incorporate the amount
of lime into the soil to adjust the pH to between 6.5 to 6.7 if necessary.
Have your soil tested by the university or other lab.  If the calcium is low
but the pH is OK, add gypsum.

Adjust the nitrogen rate to the type of tomato being grown to avoid
excessive vegetation. Use nitrogen in the form of potassium or calcium
nitrate and avoid ammonium nitrate. Apply potassium, phosphorus, and
magnesium as recommended because balancing these nutrients with calcium is
also important in preventing blossom end rot. The primary factor, however,
is maintaining uniformly adequate soil moisture throughout the season.

Calcium-rich fertilizers show mixed results in controlling this problem,
often because calcium deficiency in the soil is not the problem. If calcium
nitrate or other fertilizers are used, they should be applied to the leaves
and tomatoes when the tomatoes are about grape size. Brand names include
Tomato Saver and Blossom-End Rot Preventer. Be sure the fertilizer is meant
to be used as a leaf spray.

Other Links:
http://www.nysaes.cornell.edu/additional/agrifacts/tomato_q.a.html
(Read this for info about how to ripen green tomatoes at the end of the
season)