Every MIDI note consists of two messages, a NoteOn and a NoteOff. Each one has a note number and a channel (also a velocity, but that isn't relevant here). The MIDI standard stresses that it is invalid to send a second NoteOn for the same note number on the same channel before a NoteOff for the first NoteOn. It is more or less impossible to do this with a physical MIDI controller such as a keyboard, but remarkably easy to trigger when editing in a DAW—simply overlapping two instances of the same note will do it.

Ardour offers many options for how to deal with instances where you overlap two instances of the same note. Which one to use is a per-session property and can be modified from Session > Properties > Misc > MIDI Options.

never allow them

Edits that would create note overlaps are not allowed

don't do anything in particular

Ardour leaves overlapping notes alone—the behaviour of a MIDI receiver (plugin or hardware) is undefined

replace any overlapped existing note

When one note is moved to overlap another, remove the one that wasn't being moved

shorten the overlapped existing note

When one note is moved to overlap another, shorten the one that wasn't moved so that there is no overlap

shorten the overlapping new note

When one note is moved to overlap another, shorten the one that was moved so that there is no overlap

replace both overlapping notes with a single note

When one note is moved to overlap another, merge them both to form one (longer) note

Changing the option in use will not retroactively make changes—it will only affect new note overlaps created while the option remains chosen.

Ardour does not check for note overlaps across tracks or even across regions. If you create these, it is your responsibility to deal with the consequences.