The curve is a one-dimensional object, you can evaluate it using a single number, the curve parameter, usually called t, to obtain a point, a tangent vector, a plane… at that position (t) on the curve. You can evaluate a curve using a number within the domain of the curve, which is a property of nurbs curves, where the middle value of the domain will give you a point on the middle of the curve, and the maximum value of the domain gives you the final position of the curve. This domain can be reparametrize via right click on an input of type curve, this makes the domain of the curve from 0 to 1, so it is easier to evaluate some position using normalized values, ie, the parameter 0.5 gives you a point on the middle of the curve and the parameter 1 gives you the final position.
If you want to cut curves, you have to give the positions (in the curve dimension) where to cut. If your cutting process depends on points, then you have to calculate the parameter of the closest point using the Curve Closest Point component.