Chequered is the British spelling of checkered. The word describes a pattern of squares, light then dark, like a chessboard. A chequered flag in motor racing is the black-and-white flag that means the race is finished. People also use it about a life or a job: a chequered career means some good years and some bad years, not a smooth, clean story. Though actually, that second use is very common in news and life stories — maybe more common than talk about cloth. In American English you will almost always see checkered, with ck. Same meanings; only the letters change. It is an ordinary adjective, a little more British-newspaper than chatty, but not stiff.