Call of Duty: Modern Warfare is bringing back the single player campaign that a lot of players have missed in the previous entries in the Call of Duty franchise, but now it's back in its full glory. When you play the story mode, you obviously don't want to do the same part of the game again after you close the game for the day, so it Call of Duty: Modern Warfare?
First of all, Call of Duty: Modern Warfare autosaves very frequently. Pretty much all the time you do anything of significance, the game uses the autosave feature, so you are unlikely to actually lose any progress if you just quit the game at a random point. You can see when the game autosaves by looking at the top right corner of the screen; an icon with the text "Checkpoint" will appear every time it saves. This will happen roughly every 30 seconds or so during regular gameplay.
But how do we manually save the game in Call of Duty: Modern Warfare? Unfortunately this is not possible. The only way to save the game is by using the autosave feature, so it's simply impossible to manually force a save when you want to. This means that you can't save just before a difficult part in the mission, but must instead rely on the game to save for you. However, this is not really a problem at all, because the game autosaves so often that you won't really feel the need to manually save at any point.