Engagement involves someone having a meaningful contact or a connection. That connection normally happens when the user participate with an action. Engagement plus Trust are normally the goals of a brand marketing campaign.
That said, there are many metrics to evaluate the engagement with a brand via online campaigns. It will depend on where you look. See here same examples:
Display & Search
On display and search the metrics are the number of clicks per impression (CTR), conversions.
Rich Media Display
If you have a Rich Media Display creative you can implement many interactions that you can track. You can have a game inside the creative and capture all the interactions with the game. Frequently Advertisers opt to have a video inside the creative. Here, most of the actions measured are the video start, mute, expansion, clicks, full screens, audio mutes, completion rate and finally the CTR (clicks to impressions).
On video you can have the metric Cost per View or Completion Rate. Some advertisers also like to have as objective clicks to impreesions (CTR). In VAST 3 you can have companion ads on the video, which possibly will increase the CTR. In VPAID, you will be able to add more interactive elements, like the possibility to share your creative. You can also track this.
Facebook loves the word engagement, and it was the ppc platform that brought the word to the current popularity. Engagement is for Facebook what Quality Score is for Google. Both are based on the positive engagement of the users with the ad. Positive engagement for Facebook can be clicks, likes, shares, comments or video views. Negative engagement are post hides, spam reports and unlikes of the page. Other networks like Twitter try to follow or facebook or google.