I originally posted this on the Anime & Manga Stack Exchange site. Below is an updated and improved version for your enjoyment.
Here’s a nice diagram to visualize the School Days games as of time of writing:
School Days is the main & first game in the series. It was remade in School Days HQ with updated graphics and additional uncensored scenes. This has been released and translated into English by JAST USA. This of course is the source of the infamous anime as well.
Next in the series, Summer Days takes place during the same time period, but in an alternate universe. The story is told from the perspective of Setsuna Kiyoura. It was also remade as Shiny Days and an English translation of this has been announced recently, once again by JAST USA – and is available to preorder. There is only a short untranslated manga anthology version of Summer Days unfortunately. The original Shiny Days came bundled with two bonus games Ahoge Battle and Strip Battle Days which have not been confirmed/denied by JASTUSA yet whether or not they will be in the English version.
Whilst the main games are pure visual novels, Strip Battle Days is a game of Jan Ken Pon (Rock Paper Scissors) – and when you win, your opponent loses some clothes neutral . Ahoge Battle also deviates from the main genre and is a button-mashing fighter that not many School Days fans know about. Here’s a clip of the gameplay below.
Cross Days is a story that takes place at the same time as School Days, but with a different cast. There is no English version yet, although there was a trial CD at one stage. There is a short 2 volume manga (not translated to English yet) that covers one route as well.
Island Days, the latest installment is a Visual Novel cum Tower Defense game, where weapons picked up in the main story line are used to defend yourself during the tower-defense stage. It seems to just be a spin-off that is unplaced in the School Days timeline.
There are also untranslated light novels of all three main games, should you read Japanese.
Note: School Days has some small ties to other works by 0verflow, but they’re part of another series really.