Convolution

The simplest way to look at this is “tweaking the input so that it would look like the shape provided”. What exact tweaking is applied depends on the kernel.

Filters, kernels, weights

Those three words usually mean the same thing, unless context is clear about a different usage. Simply put, they are matrices, that are used to achieve certain effects on the image. Lets consider a simple one, 3 by 3 Scharr filter

ScharrX = [1,0,-1][1,0,-1][1,0,-1]

The filter above, when convolved with a single channel image (intensity/luminance strength), will produce a gradient in X (horizontal) direction. There is filtering that cannot be done with a kernel though, and one good example is median filter (mean is the arithmetic mean, whereas median will be the center element of a sorted array).

Derivatives

A derivative of an image is a gradient in one of two directions: x (horizontal) and y (vertical). To compute a derivative, one can use Scharr, Sobel and other gradient filters.

Curvature

The word, when used alone, will mean the curvature that would be generated if values of an image would be plotted in 3D graph. X and Z axises (which form horizontal plane) will correspond to X and Y indices of an image, and Y axis will correspond to value at that pixel. By little stretch of an imagination, filters (another names are kernels, weights) could be considered an image (or any 2D matrix). A mean filter would draw a flat plane, whereas Gaussian filter would draw a hill that gets sharper depending on it’s sigma value.