My ribs are sore after doing push ups and sit ups. Why?
Are there muscles in between my ribs? Could it be something else other than muscles that is sore, like tendons, etc?
Yes there are muscles between your ribs. They're called the intercostal muscles, though "soreness in ribs" might also be caused by soreness in your pectoralis minor (which connects to the front of the rib cage) or your serratus anterior (which connect your back to the side of your rib cage). A good hint that it's muscle soreness instead of connective tissue soreness is that muscle soreness usually feels good to massage, while tendon/ligament/joint soreness does not.
The main use of the intercostal muscles is in opposing the diaphragm in breathing, but they're used in lots of movements. I get rib soreness after doing planks for a long period, so I suspect that is the main source of the soreness (a pushup being a like plank++).
I would guess that the reason that your intercostal muscles are sore is because they are being used to hold the rib cage rigid, while forces are being transmitted between the serratus, pectoralis minor and rectus abdiminous. Dynamic tension between these muscles (and many others
*) lets your back stay flat and rigid while you're doing planks or pushups, but since a lot of the force is transmitted through the rib cage, your intercostal muscles contract and get stretched as you go through the movements. Like any other muscles after a working/stretching them, they get sore. It's probably especially noticeable since you do very few exercises which exercise your ribs.
Here's a video showing the main muscle groups used in pushups, and here's an animation showing them in action. Hopefully you can see the stresses put on the rib cage.
Pushups are really great exercises precisely because (if done properly) they strengthen so many different muscle groups (even ones you were unaware you had!). This is also one of the problems with Nautilus machines for strength training: by isolating muscles you miss out on strengthening small stabilizing muscles like the intercostal muscles and you can end up with big muscles but not genuine strength (and an increased risk of injury).
I find that stretching my rib cage can help with soreness (though stretching beforehand or immediately after doesn't do very much). For soreness from pushups, my ribs tend to get more sore on the sides of my body, and a good stretch for that is the side angle pose from yoga.
Finally, it's interesting that you get rib soreness after situps. I haven't had that experience, though situps are also complex exercises which use a lot of different muscles groups. Perhaps another answerer can go into that in more detail.
* There are a ridiculously enormous number of muscles in your back and shoulders. Planks are really complicated exercises.