The different types of Aquamarine gemstones are based primarily on the different colors and quality of Aquamarine crystals. In fact in true gemological terms (the science of gemstones) there is only one classification of Aquamarine and the various names are simply trade names and informal names.
Some of these names are based on the country or mine of origin for example Brazil Aquamarine, Madagascar Aquamarine and Santa Maria Aquamarine. Others are based on the types of inclusions such as Moss Aquamarine and Milky Aquamarine.
1. Blue Aquamarine
Aquamarine is the green-blue variety of the mineral beryl, and is usually transparent and pale in color.
Naturally occurring Aquamarine tends to be greenish-blue depending on the amount and type of of iron which causes the blue and green hues. Traditionally Aquamarine is heated to remove the green tinges to produce a purer blue color, and is a widely accepted practice. However some people nowadays prefer the original slightly green tone.
Maxixe Aquamarine is also a type of beryl with a deep blue color caused by irradiation while in the earth. However the color is unstable and fades over time once exposed to light. The stone can be irradiated manually again to produce the deep blue color, however it will simply lose its intensity again.

from New Moon Beginnings

by Metalicious
2. Brazil Aquamarine
Brazil is the primary source of aquamarine gems and Brazil or Brazilian Aquamarines are very popular and described as bluish-green.
Santa Maria Aquamarine is a mine in Brazil known for a medium-dark tone and a highly saturated blue color.
Note that often gems with a particular color but from various sources, may still be referred to by the country or mine where that color variant was first discovered.
An interesting fact I came across is that the AAA grading is a system defined by individual companies and there is actually no official standard for grading Aquamarine.

from Mystical Inceptions

from The Gem Bee
3. Moss Aquamarine
Traditional high quality Aquamarine is typically transparent with no inclusions. However as with all crystals there are deposits that will produce rough with internal inclusions due to the processes that occurred while forming. Although these are deemed lower quality, some people prefer more earthy, natural looking crystals and have become quite popular.
Moss Aquamarine is more greenish in color and has specks or threads of inclusions made up of other minerals.

from Lilys Offering

by Amy Jill Designs
4. Milky Aquamarine
Milky Aquamarine forms as an opaque or translucent gemstone. Light still goes through the stone but it is not totally transparent. These crystals tend to be pale blue in color, but are not as durable as pure Aquamarine and are usually carved into cabochons and not faceted in traditional cuts.
I could not find what causes the whitish appearance, if anyone knows please enlighten us in the comments, thank you.

from Brazil Rocks Co

by Yania Jewelry NYC
References
- International Gem Society: www.gemsociety.org/article/aquamarine-jewelry-and-gemstone-information/
- Geology.com: www.geology.com/gemstones/aquamarine/
- Gemological Institute of America: www.gia.edu/aquamarine-quality-factor
As you may know Aquamarine is the gemstone for March so these are the different varieties to choose from to gift to someone with a birthday in March.
Let us know in the comments if you have a special collection or piece of Aquamarine jewelry and what you think of these other types. *Mandy.