Eminent domain is the power of the government to take private property for public purposes. Under the Fifth Amendment to the United States Constitution, property owners are entitled to "just compensation."
Traditionally, eminent domain was used by the government to reduce transaction costs for large scale infrastructure projects, like highway or railroad construction. In mid-century New York City, Robert Moses used eminent domain to ram expressways through neighborhoods, convert 'slums' throughout the city into a housing projects, or projects like Lincoln Center, or land grants for schools like NYU and Fordham. If you take Kenneth Jackson's class on the History of the City of New York, you'll read all about him.
The 2005 Supreme Court case Kelo v. City of New London greatly expanded the scope of eminent domain powers when the majority ruled that "public benefit", such as where a city wants to take land from one private landowner and turn it over to another which it believes will generate greater tax revenue, is a "public use".
Eminent domain at Columbia is a hot-button issue with regards to its planned Manhattanville campus. While Columbia itself does not have eminent domain power, New York state does. The state controversially declared the land Columia sought to purchase "Blighted", a precursor to seizing it.
Despite the threat of state seizure (or perhaps because of it), the majority of landowners in Columbia's target area have reached an agreement with the University. Nevertheless, Columbia still benefits if the state exercises eminent domain over the entire plot, and then gave it back to Columbia, as that action would extinguish all existing leases in Columbia's buildings, freeing it of having to negotiate out of those leases before redeveloping the land.