So after some research (thanks to Laura for helping!) it looks like my tomatoes have Bottom End Rot, how original...lol.
It happens when there is a calcium deficiency. It is usually fixed by watering them. Yeah, go figure. LOL! The websites that I checked also said to discard any fruits that have been effected as SOON as you notice it. So I went out to the tomato plants and pulled off over 2 dozen tomatoes and threw them away. Now this would be the point where I would love to have a compost pile. But in my neighborhood that just can't happen. Grrrrrr. At any rate, I should be able to save the rest of the crop of tomatoes if I consistently water my plants. Tomato plants need about an inch of water each week in order to grow properly.
I hope it works so that I can at least enjoy ONE of my tomatoes.