"Go West, young man."—John Soule, popularized by Horace Greeley
Manifest Destiny was a theory and a movement in the 1800s and 1900s in America. It can be summed up as the idea that the United States of America was fated to extend from one coast to another, eventually to encompass the entirety of North America, an idea that helped fuel westward expansion. This is the secular version. A more complete story accounts for the fact that most of America was Christian at the time, and the theory of Manifest Destiny held that not only was it fate that America stretch from coast to coast, it was considered only right and good by God himself and any means necessary were to be used to make it so. They were.
As can be seen by looking at a modern map, the destiny was fulfilled, except for the part of it extending to the Arctic Ocean as well, and down into Mexico.
Though it sounded like a good idea at a time, the fact of the matter is that most of America way back when wasn't owned by the United States. America acquired it through various means, some of which have drawn criticism from later generations, such as the conquest of two thirds of Mexico and the ignoble expulsion of the native tribes from their lands and onto reservations. Even at the time, the idea was criticized by the more isolationist parts of the country as imperialistic. Among supporters, the direction of expansion, north or South of the Border, was subject to fierce debate. Northerners and abolitionists were afraid that a sea of new slave states would be carved out of the former Mexican territories, and thus demanded the Oregon country to make up for it. Arguments over which of the new territories would enter the Union as slave states and which would enter as free states were a major part of the growing North/South divide that led to the Civil War.
The phrase has also gained ground in certain Speculative Fiction works.