The traditional ukiyo-e technique of using multiple woodblocks for full color is called nishiki-e.
Nishiki-e literally means “brocade picture,” a name suggesting the rich, woven appearance of the finished prints. Instead of coloring a black-and-white print by hand, printers carved a separate block for each color and stamped the blocks successively onto the same sheet of paper. Precise registration marks, called kentō, helped keep every layer aligned.
The method was developed during the 1760s and perfected and popularized by Suzuki Harunobu. His lavishly colored prints, especially the calendar prints known as egoyomi, helped make full-color ukiyo-e commercially desirable. A single print could require many blocks, with some sources describing roughly ten to sixteen as a typical range for Edo-period production.
Aizuri-e is different: it uses blue tones, especially imported Prussian blue, rather than the full palette associated with nishiki-e. Benizuri-e used a limited scheme dominated by pink or red and green. Sumi-zuri refers to black-and-white printing, so none of those alternatives describes full-color multi-block printing.