It all depends on the form of yoga. Some styles (Ashtanga in particular, and I think Bikram as well, but I've only taken one Bikram class ever) are a set sequence of asanas (postures) which are added on to with the development of the student's practice. Ashtanga is done without a "teacher" in its pure form. You go into the studio, start the series on your own, and get adjustments from the "teacher", who is silently watching and moving around the room helping. It's very free and structured at the same time. Hard to explain. To learn the sequence, you attend a "led series", where a teacher will guide you through the postures.
Other styles can be completely varied based on what the teacher and class feels like working on that day. Some teachers will ask for requests. Most classes will have a level (beginner, moderate, advanced), and a good teacher will ask if there are beginners and give modifications/easier versions of the postures for both new students or people with injuries. Advanced classes will usually involve more inversions (poses where your feet are over your head) and may move more quickly or more slowly. Some styles are all about perfection of poses (Iyengar comes to mind) and some are more flowing (Vinyasa).
Pilates classes can vary a little, but a mat class always starts with "the hundred" and has "ab series". The exercises are very particular, and you are striving for a specific form. I've had classes called "Pilates" taught by people who have obviously never taken a Pilates class, and these are usually some version of a core-work class. Some gyms offer "yogalates" which is seems to me to be a catch all phrase for a yoga class with ab work. Regular Pilates has nothing to do with yoga. It's not spiritual, there's not a lot of stretching or holding positions. It's fast and fun and you're done.
The more you go to the classes, the more familiar you will become with it all, shootingstar. Especially if you continue with one teacher, you will soon enough learn her bag of tricks.