Any Latin American Spanish phrasebook I looked in says that the word "aqui" means "here". However, whenever I said this to anyone, for example asking a taxi driver to stop here, they seemed to correct me by saying "aqua" (or aca?). I assumed this means "there", but whenever I used "aca" people didn't seem to understand. This has been confusing me for so long, can anyone explain? What is the proper word for here, and for there, and what is "aca"?

That happened to me in Argentina. I spoke nearly no Spanish but asked where the bus stopped so I could wait for it. I pointed down and asked "Aqui" and the guard working there said, "si, si, aca!"
I assumed this use of aca meaning "here" was restricted to Argentina.
My own experience with Spanish is also limited to Argentina, but I would say that aquí is used to mean "here" when no motion is involved, as in "¡Aquí estoy! Here I am!"
Acá would be used when there is some motion involved, as in "Pase por acá. Come this way."
As to bjd's guard, my guess is that he might have been using acá because he answered as if saying "Yes, the bus will stop here," with some sense of movement in mind.
There is, of course, another possibility: that I'm just wrong in my use of the two words.
Good luck!
I just looked it up in my HarperCollins dictionary, which I think may be one of the best in print, and here's what it says:
here adv+ (+place where+) aquí; (+motion to) acá
Some of the explanations, though, don't match my experience in Argentina. The dictionary shows por aquí, por favor for "in here, please," where I would have used acá based on what I heard in Argentina. Others, though, seem to fit: for example, "Come here" is "Ven acá."
P.S. I can't help at all with "there." My dictionary devotes a couple of column-inches to the distinctions between allá, allí, and ahí, but I am convinced that I will never understand them.
Edited by NorthAmerican to add the postscript.
"Aquí, acá, allá y acullá" (meaning everywhere )
I don't know exactly how to explain it but to my native speaker ears, although both aquí and acá mean the same thing (here), acá carries a certain vagueness: while aquí is very specific and categorical, acá sounds a bit more broad and indefinite... this is one of those little language subtleties that are hard to put in words.
aca, here, aqui is also here
alla, there, alli is also there
in castellano/spanish you can speak using VOS, TU or USTED, then they can have a different/similar word for the same thing
in arg for instance aca and alla would be way more popular but not 100% just more than 90% i guess or at least in bs as, specially among some people of certain age in some parts of some neighbourhoods
in arg VOS informal and USTED formal are the most popular ways
if i go on the road i would use TU cause i would make less mistakes when speaking among people that use VOS, TU and USTED
most of the time the main difference for me in a similar word would be that the accent would be in a different letter, stronger at the begining of the word when using TU than at the end of the similar world when using VOS
also another difference is that with TU they would be more Es than As in the similar word
keep in mind that castellano/spanish specially in a country like arg is very mix cause it has been a lot of emigration
for example the CHE is very common in valencia but not all over spain
then we say ciao, chau, thats italian and so on
aquí is more-or-less "exactly here", acá is more general as palindroma says. If I want a taxi to stop right in front of my house (in northwest México), I must say, "aquí mismo".

Good thread about a subject that also had me confused = aqui vs. aca.
I was watching a Spanish movie with Spanish subtitles. The subtitle used the word "aqui" but the speaker said "aca." Needless to say, I was totally confused. If there's a difference between the two, if the speaker said "aca," shouldn't the subtitle read "aca"?

If there's a difference between the two, if the speaker said "aca," shouldn't the subtitle read "aca"?
The point of subtitles is to convey what the actors are saying but they're often not completely faithful to the dialogue (usually for reasons of space). Even if you watch an English-language film with the English subtitles on, you'll notice that liberties are taken. In addition to saving space, perhaps sometimes it's a question of the subtitler writing what they think they've heard or expected to hear.
for example the CHE is very common in valencia but not all over spain
hitchhiker, I learned my Spanish in Spain 20+ years ago but never heard the word used until I met Argentinians. Perhaps, if you came across it more recently, the Valencians picked it up from the Argentinians and not the reverse.