The main game of Star Citizen will feature a persistent online, open-world universe with people flying through it engaging in bounty hunting, trading, space combat, mining and other activities, like EVE Online and Elite: Dangerous. Star Citizen is promising a great deal more depth, however, sacrificing the accurately-rendered Milky Way galaxy of Elite (complete for 400 billion stars) for a more handcrafted universe of several hundred systems packed with missions, activities and seamless transitions from space to planet, with your character being able to jump out of your ship and engage in personal ground combat like a first-person shooter, or even jump in a ground attack vehicle.
Squadron 42 is the more focused part of the game which will replicate the single-player, story-driven drama of games like Wing Commander (Star Citizen's guiding developer, Chris Roberts, created the Wing Commander franchise back in 1990). Although personal combat, zero-gravity spacewalks and ground battles will feature, the focus will be on a military conflict which pitches your character as a fighter pilot on a large carrier vessel belonging to the United Empire of Earth. This story will be related by a large cast of characters, played by actors including Gary Oldman, Mark Strong, Gillian Anderson, Mark Hamill, John Rhys Davies and Andy Serkis. Liam Cunningham (Game of Thrones' Ser Davos Seaworth) serves as your commanding officer in the released demo. You will be able to port your character over from Squadron 42 to the full Star Citizen game.
Squadron 42 was meant for release this year but has been delayed, with no firm release date set. However, the above video shows what the full game experience should be like (hopefully with less performance issues, but that's pre-release games for you). There's another version of the video here with developer commentary.
Meanwhile, the makers of Star Citizen and Squadron 42 are being sued by CryTek over their use of the CryEngine in the game. It is unclear how serious this is and how it may affect the game's release.