Persistent Storage, in the context of Blu-ray or HD DVD, refers to memory storage to store data that needs to persist from one playback session to the next. For example, for HD DVD movies, you can add bookmarks for your favourite scenes and these will be stored in persistent storage. There are also some special features, such as Internet downloads, that will use persistent storage for downloads.
Persistent storage can be in the form of on-board memory, or add-on devices such as USB memory keys.
The HD DVD specifications specify at least 128 MB of persistent storage to be present on all hardware players. The three Blu-ray profiles (1.0, 1,1 and 2.0) require different minumum amounts of persisitent storage to be present too (1.0: 64 KB; 1.1: 256 MB; 2.0: 1 GB).