Digipacks are a type of CD packaging made out of card stock or other heavy paper/cardboard material. Digipacks can flip open like a book, or it can have three parts, so that one portion of the packaging opens to the right and one to the left, with the CD in the center portion. Usually, the portion of the digipack that hold the CD is made of plastic like a traditional jewel case CD - the plastic part is simply attached to the paper background.
Digipacks were first created by MeadWestvaco, and their product, called Digi-Pak, is trademarked. However, as the format became more popular and began to be used by more manufacturers, the generic "digipack" came to be used to describe all soft CD packaging.
Digipacks have pros and cons:
They look nice, and many bands and labels like to use them for aesthetic reasons. The three section digipack sleeves opens up more design options because there is more room. However, they're more expensive than traditional liner notes and jewel cases.
Digipacks don't crack like jewel cases do, but they will rip and eventually the paper begins to peel apart and separate.
The trays in digipacks break much more often then in jewel cases. There's not as much protection since the outer portion of made of paper, so the teeth that hold the CD in place crack and fall out easily.
When the teeth of the tray does break in a digipack, the CD falls out of the bottom of the digipack, because unlike jewel cases, there is nothing to hold it in.
Digipacks can be more environmentally friendly than jewel cases because they can be made of recycled paper - however, they aren't always in fact made in this way.