Braces in the UK are both teeth torturing devices and straps to hold trousers up.
Suspenders in the UK are clips that affix to a suspender belt, used for holding up stockings.
A garter is a circular elasticated piece of lace which goes around the leg, also to hold stockings up, or over tights, and is usually worn on a woman's wedding day or when 'dressing up' for sexual antics.
The Order of the Garter is named after the same garter actually ! "Various legends account for the origin of the Order. The most popular legend involves the "Countess of Salisbury" (either Edward's future daughter-in-law Joan of Kent or her former mother-in-law, Catherine Montacute, Countess of Salisbury). While she was dancing at a court ball at Calais, her garter is said to have slipped from her leg. When the surrounding courtiers sniggered, the king picked it up and returned it to her, exclaiming, "Honi soit qui mal y pense," ("Shamed be the person who thinks evil of it."), the phrase that has become the motto of the Order" says wikipedia.