I can speak more to the impact of the Haitian Revolution than the conflict itself.

The short version of the origins of the Revolution (which I hope someone with more expertise can get into) is that the French Revolution caused instability among its colonies and brought to the fore underlying social, economic, cultural, and political conflicts that had existed just below the surface. The conflict began as a struggle between the wealthiest white planters and lesser whites, expanded to include wealthy non-white planters, and eventually spiraled out of control to the point that slaves rose up and caused the planter regime there to implode. Metropolitan policy on happenings in its colonies weren't exactly consistent either, as the Jacobin period saw an abolition of slavery followed by its restoration under Napoleon in the early 1800s. A fairly large Napoleonic force was sent under the supervision of Leclerc to suppress the rebellion and turn the island back into a productive French planter economy, but it was decimated by disease and guerrilla warfare, returning to France a few years later as a mere shell of its former self.

The Revolution was extremely important to enslaved Africans throughout the rest of the Atlantic world. In the early 19th century, American planters were horrified by their slaves celebrating January 1st (Haitian independence day) and dressing up in uniforms similar to those worn by Haitian military and political leaders. When the Aponte Rebellion was suppressed in Cuba, Aponte's notebooks contained drawings of Haitian military officers and political leaders.

Among politicians 'another Haiti' was an oft repeated phrase. The horrors of the Revolution, which included terrible vengeance against many planters and their families, served as a rallying cry for whites in their desire to maintain order and unity. In the case of Cuba, it also served to keep white planters close to Spain despite their grievances as they feared that Madrid would rather see the island burned to ash than independent; a thought not entirely without precedent, as an Asturian Spaniard named Boves led an irregular cavalry made up of free people of color, emancipated slaves, and runaways against the first Venezuelan Republic (mostly led by white planters), which contributed to its collapse and the ruins of much of its planter economy.

Haiti was also extremely significant for Venezuela in another way. Simon Bolivar visited Haiti in the mid-1810s after another failed attempt to make Venezuela independent from Spain. President Petion of Haiti offered him arms and ships so long as he vowed to use them to help end slavery on the mainland. Bolivar was himself the son and heir of one of the wealthiest planter families in all of Venezuela, had lived his life until 1808 squandering much of that wealth, and was nursed by a black wetnurse who was likely herself a slave. Yet he accepted this support and returned to the mainland for his third attempt to make an independent Venezuelan Republic.

The failure of Napoleon's attempts to return Haiti to the status of subservient planter colony also had other impacts. Both the USA and Cuba were able to take advantage of the Revolution's disruption of the Haitian planter economy to advance their own locally produced cotton and sugar, respectively. This perhaps would have happened regardless of the Revolution, but the huge economic rewards enjoyed by both were in no small part due to a key competitor essentially dropping out of the market altogether. The fact that Haiti didn't return to its place in the Atlantic economy severely delayed the cycle of overproduction that cotton didn't hit until the late 1830s and sugar even later. Many, including Edward Baptist, have also emphasized the relationship between Napoleon's failure to recreate a booming Caribbean colonial empire and his ultimate decision to sell Louisiana to the US; a decision which was, needless to say, momentous for the future of the United States.

Haiti appears to have become both a symbol for enslaved peoples across the Atlantic world as well as a major cause of fear among whites living in slave societies. To the former it was not unlike what the USSR was in the early days to many industrial workers in the Western world, while to the latter it was similar to what the French Revolution represented to European conservatives and centrist elites.

Some reading:

Haiti and Santo Domingo, 1790–c. 1870 by Frank Moya Pons in The Cambridge History of Latin America. Volume 3: From Independence to c.1870, edited by Leslie Bethell (and not to be confused with the post-2008 collection that has a similar name!) might be a good place to start. Very much a 'textbook' introduction, it'll give you a nice summary of what you're in for without causing you to get lost in all the details.

Freedom's Mirror: Cuba and Haiti in the Age of Revolution by Ada Ferrer is a good study of Haiti and Cuba within the framework of the 'Age of Revolution'.

The Black Jacobins: Toussaint L'Ouverture and the San Domingo Revolution by CLR James is perhaps the classic text on this subject, but will not include the gargantuan amount of more recent scholarship.

The Great African Slave Revolt of 1825: Cuba and the Fight for Freedom in Matanzas by Manuel Barcia includes more relevant information on its impact on Cuba in the introductory chapter of this book.

What Hath God Wrought: The Transformation of America, 1815-1848 by Daniel Walker Howe also includes relevant information on its impact, in this case on the Early American Republic.

The Half Has Never Been Told: Slavery and the Making of American Capitalism by Edward Baptist also discusses the impact of the Haitian Revolution on American slavery.