Thanks for asking, @alexopenyoga! That's actually a pretty deep question that leads to much more...

Ashtanga means "8 limbs" (which I present here in this post); it is also called Raja yoga (raja means "royal"). Traditionally, there are only a handful of asana = physical posture routines within ashtanga yoga that progress in difficulty - primary series, 2nd series, then 3rd, 4th, 5th, and 6th series - which are classically the exact same postures performed in the exact same order every time you practice a given series.

A physical ashtanga practice usually starts with sun salutations A and B, then moves through standing, seated, inversion, and finishing postures, with some arm balances and twists thrown in the mix depending on what series you take. It is a very structured practice!

Vinyasa actually means "flow" or "transition" and can also be used to describe a defined set of poses performed in a particular order (also called a half series, which is very similar to sun salutation A). In a Western yoga class, a teacher might instruct to hold a particular pose like Warrior I for 5 breaths then "take a vinyasa" or something like that, which means to do a series of asanas in rhythm with the breath that will lead into the next pose to be held. Traditionally, a vinyasa = half series is done between every single pose of an ashtanga practice, which can make it a really great, intense cardio workout.

Ashtanga vinyasa is a hybrid style that is based on ashtanga series but with a lot more wiggle room for what postures you throw in the mix. It's rooted in the 8 limbs of ashtanga, and a physical routine might follow the same framework of sun salutations > standing > seated > inversion > finishing poses, but generally the asanas are not in the exact same order every single time. To me, ashtanga vinyasa is much more fluid than straight up ashtanga.

As far as I know, there are not other types of ashtanga, but there are many more styles of yoga. Ashtanga is actually a type of hatha yoga, which means that it is based in movement, breath, and harnessing the flow of energy through the body. Kundalini, iyengar, and bikram are just a few other types of hatha yogas.

There are also plenty of yogas that don't specifically address postures, which is really interesting to me because a lot of Westerners seem to think that twisting yourself into a pretzel is what defines yoga! Bhakti yoga is a type that is based in love and devotion. Karma yoga is the yoga of action or work. Jnana yoga is called the yoga of knowledge or of self reflection. There are even yogas of sound and of food!! These styles don't specifically address asanas.

I believe that ANYTHING you do with awareness and intention that brings your mind, body, and soul to a single point is yoga, and compartmentalizing it into different styles loses sight of the bigger picture. However, yoga is a vast field, so narrowing it down in that way can be helpful for beginning a practice.

I also believe that the practitioner is the most important person in a practice, and they are welcome to take everything they want from every single style they like to make their very own "type" of yoga. That's exactly how all these different varieties evolved over thousands of years!