MPE stands for "Multidimensional Polyphonic Expression" and is a new MIDI standard created by Roger Linn Design, ROLI, Apple, Moog, Haken Audio, Bitwig and others for communicating over MIDI between MPE controllers (like LinnStrument, Roli Seaboard, Continuum, SoundPlane or Eigenharp) and MPE sound generators (like these).
The principal reason for MPE is to get around a limitation of MIDI: Pitch Bend and Control Change messages must apply to all notes on the channel. This prevents polyphonic pitch bends and polyphonic Y-axis control (which uses Control Change messages) over a single MIDI channel. MPE solves this problem by sending each note's messages on a separate MIDI channel, rotating through a defined block of channels.
A brief summary of MPE: Each note's messages are sent on a unique MIDI channel, rotating through all channels (or a defined contiguous block of channels). The per-note messages sent over each Per-Note channel are limited to Note On, Note Off, Channel Pressure (for finger pressure), Pitch Bend (for x-axis movement) and CC74 (for Y-axis movement). Any other messages sent (like Program Change, CC7/volume, CC64/Sustain, etc.) apply to all voices, regardless of the channel over which they are sent.
In LinnStrument, MPE is called "Channel Per Note" mode.