Digital Audio Workstation (DAW)
A DAW is a software product providing a recording environment within a computer. There are many approaches to the user interface of a DAW but generally it presents a collection of tracks, where each track records a different part of a musical piece. The DAW has functions for performing the stages of a recording engineer’s workflow: defining musical instrument sounds, recording a performance of a part, editing the set of recordings and mixing the tracks into a finished product. A DAW normally contains both the functions of a MIDI sequencer and a digital audio recorder.
Direct Interface (DI)
A DI box is an electronic device that matches the electrical impedence between un-amplified instruments, such as electric guitars, and devices that operate at the so-called line level, such as DAWs, mixers, powered speakers. Without this impedance matching the audio signal passed from the instrument to the line level device will be weak or distorted. Some audio devices support switching between instrument and line level inputs so a DI box may not be required.
Musical Instrument Digital Interface (MIDI)
MIDI is a formal music industry standards specification that defines a set of messages, cables and connectors that enable encoded messages to be passed between digital audio processors. Each of these processors is essentially a computer although it might not appear so. A keyboard, called a MIDI controller, sends note-on, note-off over a MIDI cable to be recorded in a DAW as a MIDI file. An external sound source may be connected to produce reproductions of musical instruments. The MIDI interface does not transmit sound, so additional audio cables are need to route the audio produced by the sound source to speakers. The audio may also be routed into the DAW for recording as an audio file.
See the Wikipedia article https://en.wikipedia.org/wiki/MIDI for more detail.