If you want the hard definition, it's:

"Employee Engagement is a fundamental concept in the effort to understand and describe, both qualitatively and quantitatively, the nature of the relationship between an organization and its employees"

If you want the layman interpretation of that it's:

Do my staff give a shit about working here? If they do, why? If they don't, why not?

LOL. My take: proportion of the empoloyee's result that comes from their own initiative -- against that of their manager. Varies between 0 and 1.

Blessed are the companies where the middle management has no initiatives for their teams.

Forcing people to "engage" with a job they are not intrinsically motivated to do well.