Posterior core stability exercises train the body to resist excessive lumbar spine flexion (rounding). These drills include everything from the birddog all the way up through more conventional strength training exercises like deadlift variations.
Anterior core stability exercises teach the body to resist excessive lumbar spine extension (arching), and encompass a variety of drills, starting with the likes of curl-ups, prone bridges/planks, and reverse crunches. In prepared individuals, they progress all the way up through more advanced exercises like stability ball rollouts, and TRX flutters and fallouts.
Fortunately, if we understand how to classify core stability exercises, we can quick recognize that there are ways to deliver more efficient training prescriptions. Speaking broadly, you have four core stability exercise categories: anterior core stability, posterior core stability, lateral core stability, and rotary core stability.
Core stability exercises are kind of like visits to the dentist. You know you need to do them - and they keep you healthy - but they aren't really all that sexy and enjoyable. With this in mind, I think the concept of "minimum effective dose" is an especially important consideration when it comes to programming core stability exercises. We want to pick the drills that give the biggest bang for one's buck: a great training effect in only a few sets.