The acronym NGC stands for "New General Catalogue" and usually precedes a number in the catalogue, like NGC 4258 for example. The catalogue refers to objects which are extended and distinctly non-star like in nature. When the catalogue was first created the true nature of many of the objects was poorly understood, and hence there is not necessarily a very logical connection between all of the entries which range from galaxies to globular clusters, supernova remnants to nebulae.

NGC 1300, a barred spiral galaxy is an example of a typical NGC catalogue entry.
Credit: NASA, ESA

The New General Catalogue was originally compiled by J.L.E. Dreyer in 1888. There are some ~8,000 objects with NGC numbers.