Radius of curvature

So let's start with your last question, informally, the radius of curvature is a measure of how much a certain curve is pointy and has sharp corners. Given a curve $y$, you can calculate its radius of curvature using this formula:

$$\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^\dfrac{3}{2}}{\left|\dfrac{d^2y}{dx^2}\right|}$$

You might ask what radii of circles have to do with curvature, so it's worthwhile explaining it. This is a parabola:

As you can see, the sides of the parabola are pretty flat, whereas its vertex and the surrounding region (i.e.: at $x=0$) have a relatively sharp corner.

So the question is how to mathematically describe this property?

Well, one way to do it is to use circles. The part of the curve that is pretty flat can be considered to be a section of a really large circle(as shown in the picture), this circle has large radius, and hence we say this part of the curve has large radius of curvature, that is, it's very flat.

On the other hand, the vertex of the parabola and the surrounding region are relatively sharp and pointy, hence you'll notice it takes a circle with small radius to fit it on this edgy section of the parabola, we say this region has small radius of curvature.

You'll also note that, the radius of curvature for a curve, changes from one point on the curve to another, you'll further notice that, when the region is flat, the rate of change of the radius of curvature is small(you can use small number of huge circles to describe a flat region), whereas it takes a lot of circles with small radii to describe a sharp corner, and hence the rate of change of the radius of curvature is great at these regions.

How can a parabola have a center from which radius is to be measured?

No, it does not, but every point on the parabola and and the surrounding region can be regarded as a part of a circle with certain radius.

Does the radius of curvature changes with the position of body(in projectile motion)?

Yes, as stated earlier, the radius of curvature changes from point to point on a curve, since the path of the projectile can be modeled as its position on a parabola, hence the radius of curvature will change with the change of position of the projectile.