What is a catalytic domain of an enzyme?
Both parts overlap. Proteins are a chain of linked amino acids. This chain can be grouped into functional units which are called protein domains. Usually all parts of a domain are closely located in the protein and they form functional domains in the 3D structure of the protein. Proteins usually contain more than one domain (these are manifold but for example can be dimerization, activation or binding domains).

The catalytic domain is the part of the protein chain which contains the region where the catalyzed chemical reaction takes place. The 3D structure of the catalytic domain forms the active site, so the enzyme requires proper folding to be active. If you denature the enzyme, the catalytic domain will still be present (as this is a function of the protein sequence) but the active site will be gone.

As there are really many enzymes for all kinds of reactions it is hard to define a general form for the active site. A motif which is present in a lot of enzymes is the the form of a catalytic pocket or groove to which only the correct substrate of the enzyme has access. Additionally the binding of the substrate to the enzyme often induces a conformational change in the protein which closes this pocket. When the reaction is finished and the substrate is not bound anymore by the enzyme, the conformation changes again and releases the products of the reaction. See the image for a schematically drawing of this principle (image from here):