bandana
1. A species of silk or cotton handkerchief, having a uniformly dyed ground, usually of red or blue, with white figures of a circular or lozenge form, made by discharging the color. 2. A style of calico printing in which white or bright spots are produced on a red or dark ground, by discharging a color. (Webster, 1882)
His hat was pulled low over his eyes, and a red bandana was tied loosely around his neck. – By the Shores of Silver Lake, Chapter 7, "The West Begins"
A bandana is simply a large handkerchief, traditionally featuring a white pattern on a turkey red or indigo blue background. Although these were the traditional color combinations, later, purple and yellow also became popular colors. While first manufactured in India, it was the British who developed the well-known lozenge pattern (paisley) sold during the "Little House"® years. Bandanas were dyed in the converse of the way calico was made; for a bandana, the ground color was printed first, and the design area was bleached away. For calico, the darker pattern of flowers, shapes, or patterns was dyed on top of the lighter background color.
When Portuguese explorers had visited India in the 16th century, they learned about a process of dyeing material whereby the cloth was tied in knots before being dyed, causing parts of the material within the knot to remain its original color. The Hindus called this biandhnu, which through English translation of the Portuguese became known as bandanna. It is interesting to note, however, that in India in the 1870s,there was no language with a word resembling bandana in sound, which meant a kerchief, handkerchief, turban, or towel, so the word apparently came from the process used, not an item itself. In India, it was typically silk that was dyed using this process, although it was later used on cottons.
The design produced by the "tie-dying" process of knots was reproduced using lead presses by the mid-19th century, and millions of pounds of such prints were manufactured in Scotland in the early 1800s for shipment to America; they were known as "East Indian cottons." The cowboys from Little House on the Prairie and the suspicious horseman from By the Shores of Silver Lake most likely wore bandanas printed as described below: 
How bandanas were made in 1874.
The bandana is a printed handkerchief of Indian origin, usually of cotton. The cloth is first dyed in Turkey red, and then the pattern is made by removing the red color with a strong bleaching agent and a press. The pattern to be printed is cut out on two lead plates, of the full size as the handkerchief. A dozen or more are put in at the same time between the plates, and as many plates as will fill the press. The pressure is applied, and dyeing liquid is run into the upper plate, which is grooved on the underside to receive it, and holed to pass it from plate to plate through all the cloth-folds in the press. The pressure on the cloth makes clean work by preventing the spreading of the liquid dye, and is enormous: up to 500 tons is required to keep the outlines clean and crisp. The pattern in the real bandana style of printing is a series of dots and diamond prints. — William and Robert Chambers, Chambers's Encyclopædia: A Dictionary of Universal Knowledge for the People, Volume I (London: W. and R. Chambers, 1874), 658.
The bandana was an essential part of historical western gear. Its traditional dark color and pattern wouldn't show dirt like a plain white handkerchief would. It could serve - among other things - as a handkerchief, wash cloth, towel,dinner napkin, dust protector, neckerchief, pot holder, sun-screen, sling, bandage, head covering, apron, and... as a robber's face mask!
