In our product descriptions for string trick yo-yos, we throw the word "edge" around a lot. This might be new vocabulary to you, so we'll try to explain what we mean, and what your yo-yo's edge means to you.
There are many different defnitions of "edge," depending on whom you ask, but for the sake of consistency, we define edge as the point in the body's shape where the shape starts to slope towards the outer edge. Because this greatly affects the way the yo-yo handles, the height of the edge is a very important feature of a yo-yo.
High and Low Edges
Check out the photo below and you'll see what we mean. A high edge makes it easier for the string to touch the side of the yo-yo body. This can cause sleep loss, but it can also give the player an increased control over the tilt of the yo-yo. The red square marks where the "edge" is, and the black string represents the string and where on the body it hits.
On the other hand, a low edge keeps the string away from the body, minimizing sleep loss. This keeps the yo-yo spinning much longer, allowing for more complicated string tricks without affecting the spin of the yo-yo.
On the Turning Point Leviathan3, not only is there a low edge, but the body shape is stepped to keep the string even further from the side of the body.
High and Low edge pros and cons
Both high and low edges have their merits and demerits. Low edges make string tricks easier, but make looping tricks all but impossible. High edges are great for looping tricks, but are not well suited for string play. If you push the height of the edge to one extreme or another, it will always make your yo-yo play better for one style of play or another. There's a great variety of shapes and edge heights out there, which affect the yo-yo greatly. You should try them out to see which one works best for you.
There are some yo-yos that looke like they have low edges, but in fact have higher edges, due to their shape. A good example of this is the ILYY Falcon.