The Sheliak, who consider humans to be beneath them, order the removal of a human colony from a planet that, in accordance with the Federation/Sheliak treaty, they own. The colony of 15,000 was unknown to the Federation, and is made up of the descendants of a Federation vessel that crashed there more than a century ago. The Sheliak intend to colonize the planet in four days; they will likely eradicate the population if the Enterprise does not remove it.
One of the appeals of "Ensigns" is its two-tiered plot structure, in which both storylines document the problem-solving methods in an uphill climb to fix a mess of a situation before the ticking clock expires. Picard must figure out how to negotiate more time from the hopelessly obstinate Sheliak, while Data must figure out how to convince the prideful (and perhaps equally hopelessly obstinate) colonists to give up their homes and leave.
The results are mixed. This is a competent TNG story, but it has some evident problems in execution. Most notable is the depiction of the colonists in their extended dealings with Data. While Data's assignment gives him a new challenge (figuring out how to improvise while working a problem that requires extensive knowledge of human nature), a lot of these scenes simply don't work because of the belabored drama. The talky grandstanding of this kind of TNG effort requires actors that can rise to the challenge. Grainger Hines as Gosheven, the wrongheaded leader of the colony, is a wooden actor that sinks many of these scenes. In fact, a lot of the guest performances in these scenes are misfires. Data's interactions with Ard'rian (Eileen Seeley) are merely adequate.
Faring slightly better are Picard's dealings with the extremely inflexible Sheliak (whose homeworld is appropriately dubbed "Sheliak Corporate"); they continuously hang up on Picard when he tries to talk to them. Picard's bureaucratic solution to the bureaucratic problem makes for a truly funny and satisfying payoff.
Meanwhile, the scenes on the colony build to an effective demonstration of action by Data, but the ending only underlines (1) the obvious lack of communication up to that point and (2) the apparent stupidity of Gosheven and the colonists. Simply put, if the colonists know what a starship is (and they do), they should understand what kind of threat is looming without Data having to prove it.