The necessary line is an imaginary line on a football field that a team must cross to get a new set of downs.
On first down, the necessary line would be ten yards upfield from the line of scrimmage, or where the football is placed. The necessary line is roughly synonymous with the "first down marker", because announcers will usually state the ball carrier must reach the first down marker (instead of the necessary line) to reach a first down.
In actuality, the first down marker is found on the sidelines, where officials hold "the chains". The chains are actually two different tall, brightly-colored markers. These markers are held together by a 10-yard long chain, which is attached at their bottoms. One marker is placed at the initial line of scrimmage (for that set of downs). The other marker, known as the first down marker, is stretched out ten yards down the sideline, and this is used to indicate the necessary line for a new first down.
If the measurement is close to the necessary line, but cannot be determined with the naked eye, then the chains are brought onto the field for a closer measurement. In this way, officials can determine down to the inch whether the ball has traveled ten yards in the proceeding plays. A team has four plays to make 10 yards, though most teams punt the ball if that distance is not acquired in three plays. That is, they use their fourth play to punt the ball to the opposing team.
For an example of the necessary line and the first down marker, imagine a team starts their first down on their own 20 yard line. One end of the chains will be placed on the 20 yard line. The other end of the chains will be placed on their 30-yard line, ten yards upfield. This is where the first down marker will be found along the sidelines. The 30-yard line would be the necessary line.