I think I might move to Hawaii, so I guess it's still continuing. Kidding. 1848 is the common historical date for westward expansion. The movement of people westward is still occurring, as the New England population shrinks and California's population continually grows, although there are other factors involved in that as well. The west has more opportunity, both back then and now, in terms of affordable land and housing, jobs, new industries and scenic beauty. We'll see how long it lasts.
American history does not provide an exact answer to your question. Instead, there are a several fairly solid arguments that suggest a beginning and an end with regard to westward expansion. The Proclamation of 1763 proposed by Lord Grenville was an attempt to stop 'westward movement' by the colonists. It also set up a boundary line that if crossed by British colonists in their effort to move westward they would no longer have the protection of the crown. This did not stop them from moving west. The American Revolution definitely slowed westward expansion, however once the war was over combined with the new borderline of the United States (Mississippi River) people continued to move west, that was manifest destiny's beginnings. The close of westward expansion was clearly attributed to capitalist ambition. As the consumer markets leveled off in the continential United States, industry looked beyond its borders towards new consumer markets. In essence 'Manifest Destiny' by the end of the 19th century expanded to the pacific, the frontier was by all accounts 'closed'. After all, every person outside the United States was just as much a consumer as those people within U.S. borders.
The idea of expansion out west started with the Northwest Land Ordinance which gave the U.S. such states as Ohio, Indiana, and so on. But the full scale ideal of Westward Expansion started in 1803 with the Louisiana Purchase under Thomas Jefferson. Thomas Jefferson purchased the Louisiana Territory from Napoleon Bonaparte and it doubled the size of the United States. After the purchase the American government decided to use methods of Indian Removals, war, and another purchase from Great Britain to gain the Oregon territory to gain all of the continental United States that we know of today. The last piece of this puzzle was gaining the Oregon Territory which was in 1848. So in all our country completed their vision of Manifest Destiny and gained the rest of the continental territories in about 50 years! America was starting to become a super power in that time period.
The Westward Expansion of the United States began almost as soon as the colonists arrived in Plymouth and Jamestown. By the founding of the US, the concept of the "Frontier" had been established, and centered roughly on the Appalachian and Allegheny Mountain ranges, areas to the east being established states, areas to the west being unsettled areas. With the advent of the Louisiana Purchase in 1803, which doubled the size of the then United States, the number of setters increased and kept moving westward. By 1890, the "Frontier Boundary" had ceased to exist, as both coasts had been settled, and only a few interior areas remained as territories and soon became states by the first decade of the twentieth century.
The Westward Expansion really started in 1803, when Thomas Jefferson sent a secret message to Congress proposing expansion west of the Mississippi River. The idea was to send a small expedition to chart the area, but it had to be secret because France owned the land. This proposal led to the Lewis and Clark expedition.
Later that year, Napoleon of France offered the US the Louisiana Purchase for $15 million, which the US accepted. This expanded the country hugely – by about a million square miles -- west to the Rocky Mountains and North to Canada.
In 1848, Mexico ceded its rights to all land north of what is now the Mexican border, which gave the US land rights all the way to the Pacific Coast.
In 1849, prospectors found gold in California, which led the Gold Rush and a big increase in American population in the Western areas. Even without the Gold Rush, many made their way west to claim a piece of land and make their life in the new territories.