The Hungarian alphabet is simple enough to apprehend - it uses fairly familiar English letters for the most part, at least - but for people who are used to Romance or Germanic languages, some of the pronounciation may be surprising. Here, therefore, are some guidelines for pronouncing Hungarian words, letter by letter. Please note that Magyar words are pronounced, basically, completely phonetically. Any silent consonants, diphthongs, és a többi will probably only show up in "borrowed" words, and vowels are not reduced as with the a in the English word "sugar."
A phonetic representation is given for each letter, but I've tried to leave the really eggheaded linguistic terms out of here as much as possible for the sake of accessibility.
a [a]: as in "cot." The vowel in "caught" may be more accurate, although it's a subtle difference even if you pronounce the two differently at all. This is a difficult sound to get used to; in English, the letter "a" is generally pronounced much farther forward than in Magyar. I tend to think of it almost as an "aw" sound.
á [a:]: as in "baa." For some reason this is the example given in both my references. In any case, this sound is similar to the regular "a" in Magyar but much longer and more open.
b [b]: as in "bad." Same as in English.
c [ts]: as in "bits." The Magyar word cica, "cat," would be pronounced, roughly, "tsee-tsaw."
cs [t∫]: as in "chess." or "church." This one makes a lot more sense when you look at how the Hungarian s is pronounced. Also, I underlined the second ch in "church" because this sound is "soft" in Hungarian.
d [d]: as in "bid." Same as in English.
dz [dz]: as in "kids," assuming you don't actually pronounce the "s" there with any sibilance. This one's pretty much what you'd expect, actually.
dzs [dʒ]: as in "jury." Pretty much like the English "j," in fact.
e [e]: as in "pen." Same as one of e's many uses in English, anyway.
é [e:]: as in "pain," before the vowel turns into an "ee" sound. If you know French, it's like "les." Either way, this is a long vowel. Give it the length it deserves.
f [f]: as in "fit." Same as in English.
g [g]: as in "get." The g is always hard in Hungarian; it is never pronounced like the soft g in the English "mage."
gy [ɖ]: A bit of a different sound to describe, but it's similar to what happens in the middle of the two words "hid you." Think of it as a dy instead of a gy and you should be able to get by. If you're particularly careless you can get away with pronouncing it like the English j.
h [h]: As in "Hungary." Same as the English h.
i [i]: As in "sit." Same as English, but always short.
í [i:]: As in "see." Again, this is a long vowel. Don't short change it.
j [j]: As in "yellow." Don't be fooled by the phonetic symbol. This is pronounced like an English y.
k [k]: As in "kick." Nothing new here.
l [l]: As in "lip." Same as in English.
ly [j]: As in "boy." The sound is identical to the Magyar j, although it doesn't usually occur at the beginning of a word. Nevertheless because of the l in there, this one will probably take some getting used to.
m [m]: As in "woman." Same as in English.
n [n]: As in "man." Same as in English.
ny [ɲ]: As in "Enya," or Spanish "mañana." One sound; absolutely not two syllables as in English "many."
o [ɔ]: As in "tall," but shorter; imagine pronouncing the word "tal," since the double consonant l is (probably) what causes you to lengthen the vowel. The French "pomme" contains a more precise representation of this vowel, if that helps.
ó [ɔ:]: Again, roughly as in "tall," but not shortened. Also, more rounded than the vowel in this word as Americans tend to pronounce it. A more accurate analogy would be the French "beau," or perhaps the American "short."
ö [ø]: Ah, the umlaut. Actually this is not an umlaut, as you'll see if you read gn0sis's writeup in the umlaut node, but rather a symbol of diaeresis (which node you can also visit to further your knowledge, via gn0sis, on this subject), as it is in Finnish, which of course makes sense since Hungarian is a member of the Finno-Ugric family of languages, the very same family which contains Finnish! Whew! If you're not interested in all this linguistic mucking around, here's the point: An ö is a fronted "o." There's no suitable analogue to this sound in American English. A sort of close approximation would be the English pronounciation of "er," or the English hesitation-sound in which the r is, of course, not pronounced. If you know French, the Hungarian ö is similar to the vowel in the French article "le." Failing any of those, try this: purse your lips as if you were going to say the "oo" sound in "too," and then try to say "oh." The sound that results should be something like a Hungarian ö. All right!
ő [ø:]: Basically, this is the same sound as ö, but longer; like the French "deux." I'll leave it at that, but don't pronounce it just like ö! If you can't see the letter, due to browser problems, it should look like an o with two acute accents ´ side by side on top of it. This character is sometimes represented as o" on the Internet, due to sketchy support of the double-acute accent.
p [p]: As in "pi." Same as in English.
q [kw]: I don't think I've ever actually seen a q in a Magyar word, and it doesn't show up in my dictionary. I'll take metal rozsa's word that it does in fact exist in Hungarian, and assume that it's used only in borrowed words. As such, it will be pronounced as in English.
r [ɾ]: The Hungarian r is rolled slightly, not trilled. In linguistic terms I believe the sound is known as a "flap;" it is like what occurs in the middle of the American pronounciation of "butter" when it's said quickly.
s [∫]: As in "short." That's right, the Hungarian s comes with a built-in h. If it helps, you can consider the English word "sugar" instead.
sz [s]: As in "say." That's right, the only way to get rid of that built-in h is to use a z to cancel it out! So, to clarify, Magyar "s" == English "sh," Magyar "sz" == English "s."
t [t]: As in "tie." Same as in English.
ty [c]: Don't be fooled by the phoneme; this is NOT a "c" sound. Rather it is referred to as a "soft t," like in the British pronunciation of "tube" or the French "tient." A similar phoneme would be [tj].
u [u]: Sort of like in "put," but more rounded.
ú [u:]: As in "too" or "fool." Another long vowel. Don't cut it short.
ü [y]: Don't be fooled by the phoneme. This is not a "y" sound as you know it, although in a way it is slightly similar. Rather, as with the "umlaut" o above, it is a u with diaeresis, a.k.a. a fronted u. It's like the French "tu," but failing that, you can try this: purse your lips as if you were going to say the "oo" vowel in "too", and instead say the vowel sound in "tee." What results should be similar to a Hungarian ü.
ű [y:]: A long ü, like the French "rue." For non-Francophiles, just say a longer ü and that should do the trick. Again, if the character doesn't show up, it should be a u with a double-acute accent on top. If you're reading Hungarian online and you see this: u", it's probably one of these.
v [v]: As in "voice." Just like in English.
w [w]: As in "window." There are no ws in Hungarian proper; this letter is only present in borrowed words.
x [ks]: As in "tax." Same as in English; also only present in borrowed words.
z [z]: As in "lizard." Same as in English.
zs [ʒ]: As in "measure." This sound is sometimes represented in English as a "zh."
A few other things. First of all, Hungarian words are pronounced with stress on the first syllable, barring any changes in rhythm to indicate a question or other non-standard sentence. Secondly, don't swallow syllables! Hungarian affixes are very important, and so it can be vital to pronounce every syllable in a word. Finally, double consonants are actually pronounced long, unlike in English. The best way to think of this is as if the first in a pair of consonants ends one syllable and the second starts the next syllable. This will usually be the case anyway, so it can't hurt to think of it that way. For example, the double n in "honnan" (where from?) should almost be pronounced as if "hon" and "nan" were two words, but without stopping the voicing of the n between them. If a "double letter" such as sz or ty is doubled, only the first letter will be repeated: hence a long sz will be written ssz.
Many of these rules take some getting used to, but there are few, if any, exceptions to them. Now all you need to do is pick up the grammar, and you'll be on your way!
The grammar is the hard part, of course. Oh, that and the vocab. Since Hungarian is in a different language family than English and, indeed, virtually any language spoken in Europe, you're not going to find a whole lot of cognates! Have fun!
Magay Tamás and László Kiss. Magyar-Angol Szótár. Budapest: Akadémiai Kiadó, 1995.
Payne, Jerry. Colloquial Hungarian. London: Routledge & Kegan, 1987.
Thanks to Gritchka and metal rozsa for help with some of the "sound translations."