Let's start with the letters that sound just like English letters, skipping over the vowels. The second letter of the alphabet, Baa, makes the sound of a B, very simply. (It's worth noting that there isn't a P sound in Arabic, so English words spelled with a P would have the B substituted for them if they were to be written in Arabic.) taa makes a simple T sound. daal (the first letter in red) is the Arabic D. Raa is the Arabic R, and it is rolled. Zay for Z, Siin for S, Faa for F, Kaaf for K, Laam for L, miim for M, nuun for N, and haa (the one just above UU in the chart, that looks like a stick figure taking off its hat) makes the H we are used to hearing, as in "horse".
Jiim is a letter that we do have equivalents for it in English, but the sound is different depending on what region you are in. In Lebanon, for example, it makes the sound of a J, whereas in Cairo it makes the sound of a G. That is why you see the word for the Arabian Gulf (and relating dances) sometimes spelled "Khaleegy" and sometimes "Khaleejy" (or variations there of). This is also the reason we know of "Samia Gamal" (an Egyptian) and "Nadia Jamal" (whose career was mostly in Lebanon). Although in English their names are spelled differently in Arabic it is in fact the same letter, and you can find different video clips for both depending on which spelling you search.
Next, the letters we are used to hearing the sound of, but they get their own letter in Arabic: Thaa (4th letter in the alphabet) makes the sound of a "th" in the word "three" or "cathartic". The 9th letter (second one in red), dhaal, makes the sound of "th" in "other" and this is how English speaking students are taught to remember it ("it's the other one"). Take a moment to let the fact that these two totally different sounds have different letters in Arabic sink in. Shiin also falls into this category, it makes the sound of an "sh" but gets it's own letter.
Next is the Haa, you can see it written on the chart, the 6th letter down. This is the sound in "Habibi" حبيبي and in my name, "farHa" فرحة . I am writing it here with a capital H because it is not the same H we are used to using in English, although it's used for "hospital", "hurricane", and "hardly". It is a breathy sound. It is also the first letter of the word "Hob", which means "love". Try saying "Habibi" with a longing sigh, breath out and put your heart into the sound of the H. It feels good to say! You will sometimes see this letter transliterated as a 7, because of the resemblance of the number to the Arabic letter: ح As in 7abibi (which can also be written Habeebee, since the A is a short vowel and the i's are both long ^_~) Listen to Om Kalthoom sing Sert El Hob
Directly after that is the Khaa, which looks like a Haa with a dot on top of it. It is usually transliterated as "kh", and is the sound most people who have only heard Arabic shouted angrily on the news associate with the language. This is one you need to use the back of your throat for. It is in words like "Khallas" خلاص meaning "end" (over, done, finished, cut it out, salvation....) This clip of Azza Sharif she is dancing to Khalli Balak Min Zuzu (watch out for Zuzu), the first word of the chorus is spelled with this letter.
Moving down the chart we come to the emphatic letters. I wrote most the consonants in blue and these ones in black, to distinguish them. Depending on what dialect you are dealing in these letters might not sound any different than their non-emphatic counterparts, or there might be a marked difference. For the purposes of understanding different transliterations you should be aware that sometimes people will write non-emphatic letters in lowercase and emphatic ones in uppercase: such as taHTeeb تحطيب where the first letter is the normal taa, the short a is not written, then the breathy H, the emphatic T, long ii and then baa. You could transliterate this word as tahteeb, tateeb, tahtib, tahtib, tahTiib, etc. Saad, the next letter, is an emphatic S sound (found at the end of Khallas), Daad corresponds to the daal, and DHaa to the dhaal. These letters will effect the sound of the surrounding vowels.
If you go down the alphabet a little further you will see the Qaaf, which is the emphatic version of the Kaaf. This is the letter at the start of 'albi (my heart), 'amar (moon), etc. If you noticed that those words don't sound like they start with a Q it's because many dialects, including Egyptian that most our dance music is sung in, don't like this letter, and they just turn it into a glottal stop. This is why you will see the word for "dancer" transliterated in so many ways: the Arabic spelling is راقصة as in rAqaSa (Lebanese dialect will usually still pronounce that taa marbuta as an A if it comes after an emphatic letter, like the Saad ^_~). You might see raaqassa, raqessa (short vowel, pronounced differently depending on the dialect) ra'assa, rakasa, etc. You may have also heard Dina's new reality show with the title "Al RaQessa, The Bellydancer!" and then heard Saad el Shogar ask you to "ra'assni" (dance for me). It's all just variations on how to pronounce that one letter.
Lastly, the Ain and GHayn! These are both consonants, and both need the back of your throat muscles to say them. Ain is the first letter after DHaa on the chart up there written on its own it looks like this ع . You see it in words like عيني (my eye) as in "Habibi ya Aini". The easiest way I can describe it's pronunciation is that you are using the same part of your throat as the AFLAC duck. It is usually transliterated simply an an "A", but sometimes you will see it written as the number 3, thanks to the similar (but backwards) shape: as in 3ini.
The Ghayn, just below it, looks like an Ain with a dot, and it is usually transliterated as "GH". It sounds similar to the French "r" but with something of a G sound. I still make the mistake of thinking there is an R in the word when my Arabic teacher tells me a new word with the غ in it. This is the first letter of the word "ghanilli" غني لي meaning "sing for me" as in Om Kalthoom's hit song, Ghanilli Shawayy Shawayy.
! طا ظا (Ta-Da!!) Now you should know enough about the Arabic alphabet to understand why so many words have so many spellings, and what variations you can try when looking for a bangin' version of your favorite song ^_~.