1: Initial sketch on the surface of the block.
Sometimes I will plan more complicated pieces in advance, but I usually prefer to draw right on the block. I don't always stick to the sketch because I like to save some design decisions for the carving process. Before carving, the block is stained gray to improve the visibility of cut areas.

2: The white areas are carved away with tools called gouges.

The Storm, 2025, (c) Zachary Carr
3:The first test prints (or proofs) are pulled.
The rolling tool is called a brayer. The ink used is made especially for block printing and etching. It's oil based, but altered to be water soluble, similar to water soluble oil paint. For test prints, I use inexpensive paper and don't go through the trouble of centering (registering) on the paper because they are just made to check for mistakes, not for display. Here, some of the finer lines filled in with ink and didn't print, meaning they had to be carved deeper.

4:The edition is printed, numbered, titled, and signed
Once the block is ready, I print it by hand on much nicer paper, made from the long fibers of the mulberry tree's inner bark. I use a brand made for block printing. It is very thin, receptive to ink, and strong enough to stay intact during printing. I use unbleached paper because I prefer the natural off-white. The edges of prints on this paper are often torn against a ripping bar, because of how beautifully it tears.
To get a solid print, the right consistency of ink must be rolled over the block in thin, even layers. Too little will make the print patchy, too much will hide fine details.
Hand printing is done by laying paper over the inked block and burnishing it with a hard tool. There is a tool for this called a baren. I have one, but I'm on team salt shaker... the bumps on the bottom concentrate pressure just as well as those on the baren (but there is a better type of baren called a Slama press that I have not tried yet).
Printing must be done very carefully as not to shift anything, and you have to push into every little spot so the print fully transfers to the page.
Prints are easy to mess up. Messed up prints are called misprints. These aren't good enough to be included in the edition, so they are either sold for less, or destroyed.
First, the sketch. This block is hard linoleum, but any type works. Like with single colour prints, carved areas in the first layer will reveal the white of the page.

This is a "reduction linocut", where the block is destroyed in the process of progressively carving more of the block between layers. Editions of these are usually small. I only printed 9 of these. Prints with multiple layers require careful centering of each layer in the exact same place on the page.

The second layer is when it begins to get complicated. When that layer is printed with the next colour (the second layer here was red), carved lines will protect the colour of the previous layer (in this case, yellow). Each layer has to be considered carefully, either mentally or on paper, because you can't see exactly what you are doing while carving, and you are limited by a narrow range of values and colours. Another challenging aspect is that the first layer cannot be overlapped by the second layer, and so on. You have less material to work with for each layer. It works very similarly to Ukrainian easter eggs (made by adding wax lines between dye layers), which I did as a kid.
This is a 3 layer print. Before printing the 3rd layer (in dark purple), I carved everywhere that I wanted to remain red. This is the result:

...and the oldest news was printed with this very method, over 400 years ago!
Relief printing, or block printing, was the original form of printing identical text and images with ink. It likely evolved from carved seals and stamps used for clay and wax. It was invented in China as a method of patterning textiles, but it's unknown exactly when. It would have been some time before the oldest surviving block prints were created, which may have been the burial shroud and silk robe of Lady Dai (famous for the preservation of her body and her opulent burial). Some sources claim these textiles were either fully printed or made with a combination of printing and painting, including this webpage for the museum where the robe is housed. I emailed the museum for more info, and will update this if they reply. If these textiles truly are printed, then block printing existed at least as early as the time of Xin "Lady Dai" Zhui, who lived from 217 to 168 BCE.I was sadly unable to find a detailed picture of either, but her shroud and robe look like this:

If these aren't block prints, then the earliest example of printing would be one of those mentioned in this wikipedia article.
PS: the lacquerware found in her tomb is unbelievable.
The technology developed further with the invention of movable type, again in China, by Bi Sheng in the 1040s. He came up with a creative method of getting all the character pieces levelled for printing: he arranged them on heated wax and flattened them into it with a board before the wax hardened. 400 or so years later, in what is now Germany, movable type was re-invented by Johannes Gutenberg. Movable type combined with illustration blocks became the favoured way of making books for many hundreds of years, until it was rendered obsolete by newer methods somewhat recently. Block printing's practical utility survives in the form of stamps. It is still practiced today as a fine art form called printmaking, which encompasses all of the historic, handmade printing techniques.
In this timeline graph, you can see when it fell out of use, as well as the other printing techniques that developed at different points. Click on the image for the source, it's from an article that has a good overview on the different methods.
I found a cool website while researching that aggregates articles (of varying historical accuracy) on printing developments around the world, arranged in a timeline.
Click here to visit that website.
(work in progress, more sources and info will be added)