Here are brief descriptions of the main concepts behind each word. Please note that these are not "PixInsight definitions" at all. These are essential concepts of signal and image processing that are not tied to any particular application.
Structure
A set of adjacent, connected signal elements. Since we are speaking of digital images (which are two-dimensional discrete signals), the signal elements are pixels. Pixel connectivity refers to the way adjacent pixels touch each other. This also introduces the concept of pixel neighborhood, but both concepts are beyond this short description. We mainly identify an image structure by its geometry and morphology, and so we consider properties such as size (or scale), shape, orientation, and texture. We have algorithms to classify image structures as a function of these properties, including multiscale analysis, frequency domain analysis and mathematical morphology.
Scale
A range of characteristic sizes. For example, stars in a deep sky image are normally small-scale structures---unless the image consists of a single bright star. High-frequency noise is also defined at the smallest scales. Nebulae and other extended objects are larger, and we normally refer to them as medium-scale and large-scale structures. This word is more often used in the context of multiscale analysis, which comprises techniques and algorithms to classify and isolate image structures as a function of their characteristic scales.
Wavelet
A mathematical function whose properties make it suitable to perform certain types of multiscale analysis procedures (also known as multiresolution analysis). A wavelet transform uses wavelets to decompose a signal (as an image for example) into a list of components. Each component contains only structures isolated as a function of their characteristic scale (also as a function of their orientation in some types of transforms). Once we have decomposed an image this way, we can modify structures selectively at some scales of interest, and then perform an inverse wavelet transform that yields a transformed image. This is basically what the ATrousWaveletTransform tool does in PixInsight.
Layer
There are many wavelet transform algorithms. In the isotropic à trous (with holes) wavelet transform algorithm (recently known also as starlet transform), each transform component is an image of the same dimensions as the original---that's why we say that the ATWT algorithm is redundant---, and hence the word layer seems very appropriate to describe it. A wavelet layer contains image structures isolated as a function of their characteristic scale. Layers are always sorted by increasing order of scale; for example, the first layer contains the smallest structures, which are mainly noise, tiny objects and sharp edges. The multiscale median transform (MMT) algorithm is also redundant, as well as the median-wavelet hybrid algorithm (which we are implementing), and so we speak of layers in all of them.
You are right that understanding these concepts is very important to apply the related tools properly. I hope this adds more light than confusion. Let me know otherwise.
