Well... Awesome food for thought. Perhaps the largest changes would be that the British never would have made the deal for the Louisiana Purchase and by extension the purchase of Alaska. That also means most of the southwest from Texas to California would be a part of Mexico and Hawaii would be independent. Also the Civil War would have instead been a war of succession from the British Empire which would have garnered far more foreign aid and the South most probably would have won their Independence long before the US itself declared independence along with most other British holdings in the mid 20th century.
Any one of those aspects could require tons of research and days of debate to still not know the whole of what could have been.
Alaska: most probably not bought, the Russians would control Alaska and maybe parts of what is now western Canada.
Texas, Arizona, Colorado, Nevada, California and New Mexico, and surrounding areas: Probably part of Mexico or their own self-governing state.
Hawaii: Probably a British, Mexican, Russian or Japanese territory. It holds strategic value in having many harbors that can keep fleets that can sail anywhere in the Pacific, and with air carriers, it's an important strategic asset.
Civil War: Probably would not happen, because the British Empire was already starting to get rid of slaves and slaves had been made illegal in the UK early in 1807. There was a strong anti-slavery movement from the religious Christians as early as the 1760s in the UK, and a lot of countries had also abolished slavery by the early 1800s. If anything the Civil War might have been a second attempted revolutionary war.
This isn't the interesting thing. The interesting thing is the larger picture. If the American Revolution failed, the French Revolution would have probably still happened, but not with the aid of American Revolutionary generals and foreign volunteers. The Reign of Terror could have ended very differently, and there is a question of Napoleon not even rising to power. If Napoleon had risen to power, we can assume that his armies would probably still rampage over Europe, but this time the British would be able to draw from American draftees as well to fight on the continent. The question, as Moot pointed out, is Spain. Would the Spanish colonies revolt if the American Revolution failed? I think so. The British were prodding at the Rio de la Plata a few times, and the militias were the ones that drove them off, giving power to the people and embarrassing the garrisons. If the Rio de la Plata revolted, one can assume that areas with less direct contact with Spain (Chile, Peru, Paraguay), would probably revolt as well if the Rio de la Plata area succeeded. Mexico, the Caribbean and Central America is a different question. If Napoleon did take over Spain, and the River Plate area revolted earlier, Mexico might have revolted, and so would the other territories.
I got this far in when I realized why I hate "What If?"s, there's too many variables, too many strings.
History would be different, that's what, and Africa might not be suffering the aftermath of a colonial invasion and European patented pull out.