I have a rough overview on Discrete Wavelet Transform (DWT). However, I am confused about Discrete Wavelet Decomposition and did not find a good reference yet which explains this well. What is it actually about? Is it somehow part of DWT or an inverse operation to it?
The discrete wavelet transform should denote "the operations" that, applied to some data, yield a discrete wavelet decomposition. The first one can be seen as a matrix operator, while the second relates to the actual wavelet coefficients, or the structure of thereof, that you would obtain after the application of the first one.
In everyday language, they are often used interchangeably, by a form of metonymy:
a figure of speech in which a thing or concept is referred to by the name of something closely associated with that thing or concept