Our APUSH teacher last year told us that the bombs were actually duds, and that the intended damage was much more than what happened. How true is that statement? I know many nuclear bombs exist that are more explosive than the ones we dropped, but what were the dropped once intended to do?
They were not duds in any sense of the term. I don't know how people got this idea (I have seen it around), but it is utterly incorrect. The Fat Man (Nagasaki) bomb was expected to do as well as the Trinity test, around 15-20 kilotons. It did. The Little Boy (Hiroshima) bomb was expected to do around 15 kilotons, potentially less. It was around 13-15, which is an insignificant difference in terms of damage. (Damage does not scale linearly with yield.)
The Little Boy bomb was certainly an inefficient bomb — because of its design, it took 10X as much fissile material to get that 15 kilotons of explosive power as the Fat Man bomb needed for its 18 kilotons or so. But this was not a surprise to anyone involved; it was a conservative, not-very-good design that didn't work that well, and it was technically obsolete before it was even dropped.
The placing of the Nagasaki bomb was not quite correct, but as a weapon it functioned correctly.
As an aside, as early as May 1945 they had calculated that Little Boy would be between 5-15 kilotons. At that time, they thought Fat Man would likely not be better than 5 kilotons. The implosion design worked better than they expected when they tested it at Trinity.
The Bomb dropped over Nagasaki did drift somewhat as it fell, and the nearby hills lessened the damage that COULD have been done, but this not the case with the Hiroshima Bomb.
Suffice to say that aside from that, they had their intended affect; destruction and devastation the likes of which had never been seen before (EDIT) from a SINGLE bomb. Combined with the Soviet Manchurian Offensive, the bombs forced the Japanese to realize that fighting on was futile, and that unconditional surrender was their best hope.
The Fat Man design had already been tested in July, so they knew what to expect, a ~20 kiloton yield. And that's what they got at Nagasaki.
The Little Boy design was not tested beforehand, so they didn't have any hard data. Estimates were between 5 and 15 kilotons. Depending on what yield calculation you use from Hiroshima, it performed as well or slightly better than expected.
Source is the discussions about what height of burst to use, which involved estimating the yield. Note the very low Fat Man estimate pre-Trinity, so even before testing they were not expecting more than they got. Also note they were already being conservative on the height selected; had they expected a significantly higher yield these wouldn't have made any sense.
EDIT: Carey Sublette cites the interestingly precise prediction of 13.4 kilotons for Little Boy, though I don't know the original source.
EDIT 2: The book Critical Assembly contains the 13.4 kiloton prediction, attributed to Leonard Schiff.