The Red Sox are one of two teams in the American League with Sox in their name, the Chicago White Sox being the other.
The name Red Sox, chosen by owner John I. Taylor after the 1907 season, refers to the red hose in the team uniform beginning 1908.
Actually, Sox was adopted by newspapers needing a headline-friendly form of Stockings, as "Stockings Win!" in large type would not fit on a page.
In 1901, the American League declared itself equal to the National League and established a competing club in Boston. For seven seasons, the AL team wore dark blue stockings and had no official nickname. They were simply "Boston" or "the Bostons"; or the "Americans" or "Boston Americans" as in "American Leaguers", Boston being a two-team city. Their 1901-1907 jerseys, both home and road, simply read "Boston", except for 1902 when they sported large letters "B" and "A" denoting "Boston" and "American". On December 18, 1907, Taylor announced that the club had officially adopted red as its new team color.
One of the American League's eight charter franchises, the Chicago team was established as a major league baseball club in 1901. The club was originally called the Chicago White Stockings, after the nickname abandoned by the Cubs, and the name was soon shortened to Chicago White Sox.
Hope this helps with an understanding of the Red Sox and White Sox names.