The latest sunrise of the year will occur on January 5 this year...2 full weeks after the shortest day! The earliest sunset occurred on December 8 even as the days continued to get shorter and on December 21-22 (the Winter Solstice) we experienced the shortest daylight period of the year. Since then the days have been getting longer, even as the sunrise was getting later.
It might seem as if the latest sunrise and earliest sunset should occur on the shortest day, but both the tilt of the earth's axis and it's slightly elliptical orbit work together to speed and slow the sun relative to our clocks, sometimes pushing the daylight period later into the day (as has been happening in the last month), and other times moving the daylight period into the morning in a predictable pattern we call "the equation of time".
The term solstice means sun stops, or sun stands still. Of course the sun is always moving east to west across our sky, but from late November through mid January, the sun is nearly as far south as gets (it stops moving further south!) - and that's why we see such uniformity in the length of the day....9 hours 20 minutes on 12/8, 9 hours 13 minutes on the solstice, and 9 hours 20 minutes again on 1/4. It isn't until late February that you'll really notice rapid lengthening of the day.