G'MIC Reference documentation assumes knowledge as it was written by scientists for scientists.
This page is intended to make it more user friendly.
Many effects require finding the "best" result for small patches all at the same time, technically that is a Partial Differential Equation (PDE), an equation about how each pixel should look relative to its neighbours. This equation is the same everywhere so looks nice and simple mathematically but to solve all of those small patch equations at the same time you need to start at the edges and work inwards. Boundary conditions are the bits at the edge which do not change. Often in GMIC that means around the frame of the image.
What do the options actually do:
The GIMP plug-in has a filter Various / Custom Code. You can find a function then experiment with the different settings and see them in action.
Images serve many purposes in G'MIC, not all them pictorial. The Handbook will frequently make reference to a type of image that is produced by a command, or is to be used as a command argument, without much of a hint on what particular features distinguishes that image type from others, or how the components of an image are to be interpreted by that command. Here is a cheat sheet of some of the non-pictoral jargon associated with so-called 'images':
vector field: width (x) and height (y) locate a pixel; the channels furnish the dimensional components of a vector rooted at that pixel. By convention, channel 0 holds the x (width) component, channel 1 the y (height) component and channel 2 the z (depth) component. Vector fields are not limited to three channels, or components, but two or three component vector fields serve almost every purpose in G'MIC. Two dimensional vector fields appear as red, yellow and green images under the G'MIC -display command, as it interprets vector fields as two channel RG images.
displacement field: Commonly seen around -warp and -displacement, the vector field indicates how far and in what direction a pixel has been moved (-displacement) or should be moved (-warp in relative mode) or absolutely placed (-warp in absolute mode). Also called a warping field.
orientation field: Commonly seen around -orientation, a vector field where all of the vectors are of unit length; the significant datum associated with each vector element is the direction it indicates, as all lengths have been normalized to unity, hence the term 'orientation'.
tensor field: width (x), height (y) and possibly depth (z) locate a pixel; the channels furnish the diagonal and upper off-diagonal coefficients of a matrix, or tensor (two names for the same thing) associated with the pixel at coordinates (x,y) or (x,y,z) The tensor in question characterizes the gradient field as 'sampled' at pixel (x,y) or (x,y,z) and is symmetric. Given this, it is only necessary to store three coefficients (for 2x2 tensors) or six coefficients (for 3x3 tensors), as the others may be inferred. The 2x2 tensor uses three channels to store tensor coefficients xx (channel 0) xy (channel 1) and yy (channel 2). The 3D tensor requires six channels to store coefficients xx (channel 0) xy (channel 1) xz (channel 2) yy (channel 3) yz (channel 4) and zz (channel 5). Generally seen around -smooth and anisotropic smoothing friends (-structuretensors, -diffusiontensors, -eigen2tensor).
matrix field: Alias for tensor field.