Some of these terms are new in our culture, particularly transmedia. Each one is similar to the other, but different. So I thought I would write an article to clearly explain the difference between the three terms, particularly because this site is built surrounding each of these types of media. So here it goes.
Multimedia:(Note: Multimedia can refer to communications technology, but in this case refers to it's use in entertainment.) Multimedia in this case refers to one story that is adapted and converted into multiple mediums. This happens all the time with books and film. For example, the Harry Potter books were each adapted to a film, a video game, and other mediums. However, they all kept the same basic story, taken from the book.
Crossmedia: Crossmedia is when you get a series of stories that each take place in the same world. These are more and more common these days (I have multiple lists here on this site) and were spearheaded by franchises like Star Wars, Halo, etc. These separate stories are often self contained within the individual story, and can make use of multiple mediums like novels, film, video games, comic books, etc. However, though they are interrelated, they are not all part of the exact same story, just the same universe.
Transmedia: Transmedia is a fairly new form of storytelling. Similar to crossmedia they make use of multiple media to tell the story. However, different from crossmedia, transmedia uses multiple mediums to tell the same story. Think of it as a novel where each chapter is a different medium. These often include an interactive online element, though this is not a requirement. For example, the department store Target recently started a web series called Falling for You which is about two fashion designers that fall in a relationship but are also competitive in their design. Viewers can vote on who wins, and the winners will be part of Target's fashion line for the next quarter, thus providing an interactive element and using multiple forms of media to engage the audience.