MOSCOW—A suicide bomber struck in the southwest Russian city of Volgograd on Monday, killing at least 16 people in the city's second terrorist attack in less than 24 hours and stoking security fears in the country ahead of the Winter Olympics.

Russian President Vladimir Putin ordered security to be tightened around the country after the second attack, which comes less than six weeks before the games are set to start in the Black Sea resort of Sochi. The repeated bombings, and risk of additional violence, threaten to tarnish...