Organic throws are made of wood, and inorganic throws are metal or plastic. 
Really should be, anyway…
I took this from another store site, and removed the links to the store. Hopefully that’s okay? The shapes they call “Round” and “Step-Round” are the ones that are being called “Organic” here:
About Yo-Yo Shapes
Yo-Yos come in many different shapes and sizes, which make them effective for particular styles of play. The shape affects what kind of tricks you can do, so please bear this in mind when making a purchase.
For Looping Play
Imperial Shape
Stood on its side, the yo-yo has an elliptical shape, with a thin gap bisecting the round shape. This is the oldest and most traditional yo-yo shape, and is what most people picture when they imagine yo-yos. The shape itself is named for the Duncan Imperial Yo-Yo, which popularized the shape.
Despite its irregular stepped shape, the Hyper Yo-Yo Windorbit is still technically an Imperial Shape.
Raider Shape
This is the most popular shape for looping yo-yos today. The sides are mostly symmetrical, forming two ellipses side by side. Compared to the Imperial Shape, Raiders can also be used for some string tricks, making them good all-around yo-yos.
For String Play
For string play, most yo-yos are what’s called Butterfly shape (named for the Duncan Butterfly yo-yo). This shape is wide, giving you a large catch zone to land the yo-yo on the string. This makes even simple tricks like Elevator that much easier to execute. All of the yo-yos you see below are Butterfly shape; if nothing else, this introduction will show you just how much variation there is within a single yo-yo shape.
Round Shape
The simplest name for the simplest shape, it fits easily into your hand.
Pictured: Duncan FH Zero
Straight Shape
Another straightforward name, for the straight line drawn from the response to the outer rim. Straight bodies do a better job of keeping the string from rubbing the yo-yo, causing sleep loss.
Pictured: YoYoJam PHENOMizm
Step-Round Shape
Looking from the center, the straight shape suddenly “steps” into a rounded rim. This combines the comfort of round with the sleep loss benefits of straight yo-yos.
Pictured: Turning Point Leviathan2
Step-Straight Shape
The straight shape takes a new angle, stepping into a polygonal second profile line. This helps push weight outwards for various reasons.
Pictured: yoyorecreation Sleipnir
H-Profile
This shape has a very distinct stepping, and is easy to spot in a crowd. When you hold it, your middle finger will naturally fall into the gap, giving it a unique feeling favored by counterweight players.
Pictured: YoYoFactory Genesis Throwback
Inverse-Round Shape
A new concept opposite the traditional round shape, sleep loss is reduced greatly as the string only comes into contact with a point on the body, rather than an entire face.
Pictured: YoYoJoker DoubleJoker
Flat Rim Shape
Flat rims are described as rims wider than 4mm that lay parallel to the axle. This unique shape cuts into the trapeze width (catch zone), but it gives impressive spin to the yo-yo.
Pictured: onedrop Markmont Next
Pictured: YoYoFactory MVP
A Word on Shapes
Yo-Yo names and shapes are far from standardized, every place may have its own vocabulary to describe yo-yos, and makers have their own terminology as well. This page is meant just to introduce you to the variety of yo-yo shapes out there, and not to preach to you one name over another. However, we try to stay consistent in the way we refer to yo-yos here at ******.