You can play the spec-ops or multiplayer game modes in local two-player split-screen. You are unable to play Call of Duty: Modern Warfare's campaign in split-screen, or play with more than two people on one screen.
The spec-ops game mode, often known as the co-op game mode, can be played in two-player local split screen without limitations. The multiplayer split-screen applies some limits on what multiplayer game modes can be played. Common game modes like Team Deathmatch or Domination are available, but alternate game modes such as Infected or Free for All are prohibited as having two players working together may provide an unfair advantage.
You can also play local multiplayer called "private matches" which pits you and your ally against each other or against bots using maps and difficulties you select. These private matches do not allot any XP to your multiplayer characters but are heavily customizable and can be played offline in split-screen.