Boston's red shirt is their alternative jersey. It is worn for a few different reasons. One is to honor certain days that are important to the program.
Over the past few years teams have started to use third and sometimes fourth uniforms. The use of these uniforms depends on the teams. SOme use these uniforms just on sundays.
The Boston Red Sox wear red shirts on afternoon games during the week. They also can wear them on Sunday afternoon games, usually at the end of a homestand. They can also wear them for special occasions.