"kind of woolen fabric," mid-15c., perhaps from French tiretaine "strong, coarse fabric" (mid-13c.), from Old French tiret "kind of cloth," from tire "silk cloth," from Medieval Latin tyrius "cloth from Tyre" (see Tyrian).
If this is the source, spelling likely influenced in Middle English by tartaryn "rich silk cloth" (mid-14c.), from Old French tartarin "Tartar cloth," from Tartare "Tartar," the Central Asian people (see Tartar). Specific meaning "woolen or worsted cloth woven with crossing stripes of colors" is from c. 1500, formerly a part of the distinctive dress of Scottish Highlanders, each clan having its particular pattern.
1510s, "garment consisting of a long piece of woolen cloth, often having a tartan pattern, traditionally worn in Scotland," from Scottish, from or related to Gaelic plaide "blanket, mantle," a word of uncertain etymology, perhaps a contraction of peallaid "sheepskin," from peall "skin," from Latin pellis (but OED finds this "phonetically improbable").
It is a large rectangular piece of woolen stuff, and is worn in Scotland by both sexes for warmth and for protection against the weather. It is a special dress of the Highlanders, and forms part of the uniform of certain infantry regiments in the British army. A variety of the plaid is called maud. [Century Dictionary]
The wearing of it by males was forbidden by Act of Parliament, under penalty of transportation, 1746-82. The meaning "a pattern of bars crossing each other at right angles" is by 1890. As an adjective, "ornamented with a pattern of bars or stripes of color crossing one another at right angles," c. 1600, from the noun.