The Training Machine is an item from Super Paper Mario. It is a 2-screened handheld that resembles a Nintendo DS. Mario received it from Bestovius to deliver to Merluvlee, after giving Bestovius the "You-Know-What".
Upon giving it to Merluvlee, Mario questions which body part she would use it to train, and she cheerfully replies that she trains her brain, which is a reference to Brain Age. She also says she has trained every other part of her body, which is a reference to Wii.
Apparently, the Training Machine is an old model; Bestovius claims to have a newer, slimmer version (possibly a Nintendo DS Lite).