If you want a proper scientific/medical response, I'd start Googling around for medical journal articles and the like. There's plenty of information on the benefits of cardio out there. In short, it strengthens your cardiovascular system and improves your stamina. Here's a link to get your started:
http://www.mayoclinic.org/healthy-li...-20045541?pg=2
For me, cardio makes me feel good and the more cardio I do, the better I get at it. So I run in order to feel good after, get outside, and improve my stamina for activities I enjoy like dance and mountain hiking.
I'd count pushups and sit-ups as bodyweight strength exercises. Useful, but not cardio. Cardio gets your heart rate up to higher levels.
When you lift weights and do cardio depends on your goals. Check out this recent thread where the topic was discussed:
http://www.3fatchicks.com/forum/exer...-training.html
In my opinion, the best exercise program is one you enjoy and can stick with. Good luck finding which activities you enjoy most that help you reach your goals!