A brief outline.
Woodcut and wood engraving are both 'relief' printmaking techniques. For woodcut, the wood used is the plank part, just like your coffee table is, the tree trunk or branch is cut longways, ( really sorry, trees ). Wood engraving is using the end grain of the wood, think of a branch cut like a loaf of bread. The tools used to cut, or engrave the wood are really quite different. Wood engraving requires finer, more pointed tools, and woodcut tools are more like carving tools, 'u' shaped and 'v' shaped. There are some engraving tools shown in the 'Wood and Tools' section of this website.
Wood engraving blocks tend to be much smaller than wood cut and the surface is very smooth to make cutting easier. Typical wood for engraving is boxwood, but other slow growing trees can be used too, like Apple, or Holly. A leather pad is used to turn the block while cutting.
To print the woodblocks, once carved or engraved, ink ( oil or water based) is rolled or scraped onto the surface. Then depending on preference, or access, the back of a wooden spoon or a printing press is used to take the impression(s). Multiple blocks can be used for colour prints, or, as I have used in the past, the block can be printed using the 'reduction' technique where only part of the block is cut away, and a light colour printed as a background or perhaps easier to explain as an undercoat. Each subsequent colour is printed having cut away more of the block, with careful registration to get a clear image if desired.
I now use oil and water based inks, for my wood engravings, so that some of the ink goes into the incised ( engraved ) areas, and I am still experimenting with that very different and non-traditional way of printing my blocks.
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Thanks for reading this not so brief, and by no means a definitive description of woocut and wood engraving. there is plenty of info online which covers these printmaking techniques in great detail.