Plaid: refers to the pattern of crisscrossing lines, usually of varying widths and colors rather than a simple grid. The word has its origins in the Scottish Gaelic word for “blanket.”
Tartan: a subset of plaid, associated with Scotland. A tartan plaid is made of perfect squares; it looks the same rotated 90 degrees, 180 degrees, etc. (I learned this from a CBS News Sunday Morning story here.)
Different tartans are not only associated with different Scottish clans, but with all kinds of places, institutions, and events. You can search for registered tartans alphabetically, or by name via the Scottish Tartan Authority‘s “Tartan Ferret” search. The term refers to the pattern only; it’s still tartan whether it’s a woven fabric, printed fabric, or a print on a biscuit tin. A more detailed, technical definition here.
Madras: Refers to the lightweight yarn-dyed woven cotton fabric from Chennai (formerly Madras), India. It can be any design — usually plaid, checks, or stripes — but in the U.S. the term is more associated with summery plaid designs, and are often found as pre-sewn patchwork fabric. (Thanks to Kim of Spoonflower for asking about Madras plaid!)