Of course, there were a large number of revolutions or rebellions against Spanish (and Portuguese) rule during this time. It is not really possible to summarize each of them here, but we can discuss the major factor that was common to all of them.
The Latin American rebellions were about criollos wanting more power than they were allowed to have by the peninsulares. The criollos were people of Spanish (or Portuguese) descent who were born in America. These were people who had the same ethnic background as the peninsulares (those actually born in Europe) but were discriminated against nonetheless. (Please note well--this was not a case of Indian or part Indian people rising up for equal rights for all. This was a case of some people of European descent wanting to have their "rightful" place at the top of society while still keeping the Indians and mestizos below them.)
The criollos felt that this was extremely unfair. They felt that they were just as good as the peninsulares and felt they should have equal rights. These feelings came to a head when Napoleon conquered Spain and Portugal in the early 1800s. This led the criollos to question whether they should still have any loyalty to a "mother country" that was not even really controlled by their own countrymen.
Therefore, the revolutions came about both because of Napoleon's conquest of Spain and Portugal and because of the criollos' desire for more power.