There were indeed advertisements, but each place operated differently in that regard. In the United States, I don't know of any ads placed directly by the federal government, but state governments (http://www.wisconsinhistory.org/turningpoints/tp-018/?action=more_essay) did. For the USA, South Africa, Canada, and New Zealand (the 'white' dominions as it were) there was also advertisements in Europe (and especially Britain) targeted at attracting people to the new colonies. More 'colonial' states, like India, and certain African Colonies, women were also specifically targeted (which was nothing new; similar strategies had taken place for Quebec and New England).

These new colonies were not the only ones advertising of course. There was also a big influence from shipping companies, who stood to benefit from the increased traffic (and indeed, some countries placed the blame squarely on shipping companies for attracting their poor workers). http://www.russborough.com/antique_prints/posters/canada_immigration_posters.html#white_star_olympic

As for the decision why, it is almost certainly a factor of opportunity. For example, let's take Canada and the USA. Canada did not really experience an immigration boom until land in the United States became more scarce. The United States, Canada, Australia, and many other colonial states offered the chance at owning land, something that we take for granted here in North America, but which was extreme rare in Europe during this period. Arable land continued to be controlled in the majority by wealthy estate owners, farmed by tenants. These vast new places not only offered the chance at ownership, they offered it for essentially the price of a steamship ticket. Other reasons people were attracted to particular countries stems largely from nationalism, language, and politics. The United States was very attractive to certain people seeking a more democratic environment (Germans and Irish were particularly keen, for obvious reason) while the colonial empires provided a supported environment for those who wished to remain under the control of their home countries. Some instances seem a bit confusing here as well. For example, why did Argentina receive so many immigrants while places like Colombia or Peru did not? The answer is the land issue again; the Rio de Plata and surrounds is much closer to the land characteristics of North America than what we associate with South America.

Canada actually had an entire campaign to lure immigrants to the Prairies. In (what is today) Saskatchewan (and parts of Manitoba and what would become Alberta) there were vast swaths of land that were going unoccupied and unfarmed. In order to lure European immigrants, the government of Canada began the "Last Best West" campaign, advertising the Prairies as the last great plains left to settle.

The Dominion Lands Act also gave newly arrived farmers 160 acres of land to cultivate, for only a small administrative fee, a further incentive to move to Canada.