In a transporter system, the matter stream referred to the energized form of the matter being transported. The term was also used to refer to the region of normal space within a Borg transwarp conduit.
Transporter matter stream
The matter stream consisted of the stream of sub-atomic particles that resulted from the dematerialization of a transport subject. The stream was relocated from the original site of the subject, passed through the pattern buffer, transferred as an energy beam, and rematerialized into its original form again at its destination (often a transporter platform).
Although some transporter models allowed motion within the matter stream, it was advisable to remain still. Additionally, normal spatial relationships within the stream were often distorted, making objects appear differently then their actual proportions.
The matter stream was contained in the pattern buffer before rematerialization occurred. If a person was in the matter stream too long, his or her pattern would degrade to the point that rematerialization was no longer possible. Therefore, the signal resolution needed to be kept above fifty percent. (TNG: "Realm of Fear", "Encounter at Farpoint")
When the object to be transported was surrounded with victurium alloy--for example, if a person was lying under a collapsed bulkhead--resolving the matter stream was not possible and the person needed to be moved to another location or the victurium needed to be removed. (TNG: "Hero Worship")
Transwarp conduit matter stream
The area of normal space within a Borg transwarp conduit, known as the matter stream, had the effect of accelerating anything entering it to move forward.
In 2375, after the Delta Flyer had exited a conduit, the USS Voyager fired a spread of photon torpedoes into the threshold perimeter in hopes of destabilizing the matter stream within and imploding the conduit with a pursuing Borg ship still inside. The conduit did not implode and collapse, but the Borg ship was blown to pieces due to the destabilization of the stream. (TNG: "Descent"; VOY: "Dark Frontier")