This is a term used in many sports, football included, which denotes a game played away from the team's home stadium. A road game is also known as an "away game." In the NFL, every team plays the same number of home and away games, eight of each. The league typically tries to stagger it so as not to give any team an advantage or disadvantage in regards to repeated home or away games for an extended period.
When preparing for a game on the road, a professional NFL team will typically practice all week and then leave for the game on Friday or Saturday. A game on the road is considered more difficult because the team is presumed to not be in their comfort zone since they are away from home. Before the modernization of air travel, teams would travel by bus, which is why they are considered "road" games. In many semi-professional and minor leagues, teams still travel by bus because air travel is too expensive. As one might expect, travel by bus is typically a harder trip, so a road game did (and still may) entail a bit of suffering on the part of the away team.