An item's 'radius' refers to how much it is curved. When you've heard it used, it was likely referring to the fingerboard of a guitar, but there are other important parts (the bridge saddle height, for example) where radius becomes important.
In mathematical terms, a radius is a linear measurement from the centre of a circle to the perimeter of the circle. However, when you're talking about something like the guitar's fingerboard, obviously it's not shaped like a whole circle, but it IS curved, so that means it's like taking a small piece of the outside of the circle.
Imagine this: instead of a flat piece of wood, your fingerboard starts as a big cylinder, and we just cut off a piece from the outside to make the fingerboard. If the tube has a 12" radius, that means from the edge of the tube to the centre, it's a foot! And another foot to get to the opposite edge! So your tube is really 2" wide at its widest point!
Then you take a shaving from that and you end up with a fingerboard that has a 12" radius.
On the other hand, if your wooden cylinder has a 15" radius (it's even bigger!), when you take off a strip from the outside that's the same width, the remaining curve is going to be flatter, right?
A picture would explain it better, but I couldn't be arsed right now. Hope that helps.
Is it necessary? Not strictly speaking, no. But it'll be more comfortable to play. If I'm not mistaken, classical guitar fingerboards actually have a 'negative' radius, so they're curved in instead of out. But I haven't touched a classical in so many years that my memory might be funny.
For a solidbody electric, I can't imagine not having a radiused fingerboard.
Greg