Pilates was developed by Joseph Pilates, and his own discoveries as to what made a better athlete.
The work in Pilates is centered around the body core -- in other words, the stomach and the back -- and attempting to use these muscle to move your arms and legs.
The idea behind that is that you are more efficient and can move better with that strong core strength.
J. Pilates first did his exercises based on runners on other track and field athletes, and later it was incorporated by ballet dancers (and other dancers) who found that Pilates (the exercise) was excellent at improving muscle stregnth without bulkiness.
As you know, dancer don't need the bulkiness that other atheletes may be looking for.
Pilates, as an exercise, makes you aware of how your abdominal muscles are used in every day life -- from picking up the trash, to walking down to your car, to how you carry yourself (posture improves greatly!).
Pilates is also great for people who have had injuries, as most Pilates instructors (at least those worth their while) have had to deal with physical therapy cases. People with back pain or injuries have found that Pilates helps those injuries, people with knee surgery have also found that Pilates helps stregnthen muscles that take the pressure off joints, etc.
There are a number of reasons for doing Pilates, and I have been doing it for the last 8 years. It's an absolutely amazing exercise.
If you start off with the mat classes (literally, laying on a mat) those classes will focus on your abdonimal core, and possibily flexibility.
If you decide to take duos/trios or individual classes, those classes will take you on the machines -- there are a number of different machines, the more well known being the Reformer and the Cadillac. The Reformer uses springs as a way to stregnthen and lengthen those muscles. Both machines will require you to build upper body strength and flexibility through movement.
There is NO philosophical component to Pilates.
I took Yoga for many years, and yoga does have a meditative aspect. There are different types of yoga, so you'll have to find out which ones you want to take. There is the more "religious/meditative" yoga which is great if you are a really stressed out person or looking for that inner peace.
Then you have the other extreme which can be either a Pilates/yoga combination, which is more cardio oriented or the Birkham yoga which is definitely meant to make you lose 10 lbs (I exaggerate) of water weight (classes are kept very, very, warm).
Either way, both classes overlap in some positions and exercises, but Yoga has a decisively spiritual aspect that Pilates does not. Yoga was created as a means to reach that perfect meditative state in which to find inner peace and god (in Eastern religion).
Anyway, this is pretty long, so I'll end now, if you have any other questions about Pilates, please let me know! I'd love to answer them!