In football, 3rd and goal refers to the downs system. A down is one chance for a team's offense to run a play in an attempt to advance the ball. A team has four downs in a series to reach the first down line, which is ten yards downfield from where the first snap in the series took place.
Describing a team's position on the field consists of the down the offense is on and the number of yards the offense needs to reach the line to gain. If a team is on its third down and has gained six yards, it would be referred to as third and 4 in reference to the four yards between the line of scrimmage and the line to gain, which is often called the first down line.
However, if the goal line is within ten yards of the first snap in the series, the number of yards between the line of scrimmage and the goal line may be replaced with "and goal." On third down in this scenario, field position can be described as third and goal, regardless of how many yards are necessary to reach the end zone.
This applies to first, second, and fourth downs as well. In other words, a play can be described as "first and goal," "second and goal," and "fourth and goal." A third and goal situation is unique in a sense because it is the last play in a series of downs where the offense will almost certainly still attempt to score a touchdown. On fourth and goal, the offense might still try to score a touchdown but might also elect to kick a field goal instead. Of course, the offense might punt too, but only if they somehow got out of field goal range.