In layman's terms, Cosmic Awareness is a term used to describe the ability to know all. This does not mean that they do in fact know everything, it means that they have the potential to know everything, in that they have the mental capacity to remember all knowledge they come into contact with. This is the ability to be able to retain all available knowledge presented to a person, collective, or entity. Synonyms for this ability are "omniscience", "all-knowing", and "pansophical".
Perhaps the most prominent examples of those with cosmic awareness are the Guardians of the Universe, who have the ability to retain all knowledge they have ever collected without losing them by natural means. This allows them to remember all past deeds and exploits, to remember all languages they have come into contact with, and to gain new knowledge without detriment to past memories. It is their knowledge that the Oan central computer draws it information from. These memories can however, be stolen, as Ganthet claimed that Parallax stole his, and the other Guardian's memories of him so they would not stop his escape from the Oan Central Power Battery.