Hi sunshinecat, and welcome to Scubaboard!! Congtratulations on completing your Open Water Certification and also for looking to further your training and experience.
I'll try to break down the certification levels for you:
Adventure Dive: a dive with an Instructor that introduces you to something new, such as Peak Performance Buoyancy, Underwater Photographer or Night Dive.
3 Adventure Dives= Adventure Diver
3 Adventure Dives + 2 additional dives (deep and navigation are required) = Advanced Open Water Diver
Many Specialties can be completed with your existing Open Water Certification. Some others, such as Deep Diver and Wreck Diver, can only be completed with an Adventure Diver rating. Still others, such as Ice Diver and Search and Recovery Diver, can only be completed with an Advanced Open Water rating.
The first dives of most Specialties can, at the Instructor's discretion, be counted as Adventure dives towards an Advanced Open Water rating.
PADI gives a lot of ways to increase your training. AOW is a great next step where you will be introduced to new ways to explore underwater. Specialties provide you with in-depth training in areas that you find interesting. Dives can, in many cases, be counted different ways that fit where you want to go.
If this doesn't help clear things up for you
then there is a lot of info on the PADI website as well as being able to contact PADI with specific questions.
Enjoy your new adventures and keep exploring the options that are out there as to what interests you.
Hope this all helps,
Hank