I would like to know what letters and sounds these numbers represent when phonetically writing Arabic with the English alphabet. For example in these lyrics from Mohamed Amaki's
I can guess tha 7 represents Ah (what letter is this?)

Ahla Haga Feeki:

Mohamed Hamaki - 3arfa A7la HagaFeeky Eih (do you know what's the best thing about you)

Bet7ly Ay Shee2 3eneeky Teegy Feeh / You make eveything that you'r eyes look at beautiful

I am starting to learn more Arabic words and I am excited about that. From previously studying Spanish I have learned that songs are a great way to learn vocabulary and get some understanding of pronunciation/accents :) Thanks for your help!

First, the numbers are only used in informal Arabic (usually for chatting on the Web).

If you know how the name "Ali" is pronounced in Arabic, the first letter's pronunciation is usually refered to as "3". In Arabic, it's "?" (ayen). In the IPA, it's called "voiced pharyngeal fricative". So it can be written as "3li".
See: http://en.wikipedia.org/wiki/Voiced_pharyngeal_fricative

Also, "7" refers to the Arabic pronunciation of the "h" as in the name "Mohammed". It's "?" (ha'), and the sound is called "voiceless pharyngeal fricative". So we can write it as "Mo7ammad".
See: http://en.wikipedia.org/wiki/Voiceless_pharyngeal_fricative

In the word "su'al" (???? = question), the apostrophe (short stop) is usually replaced by "2", so we can say "su2al".

7 soinds ''h'' but from the bottom of your throat

3 is (ain) but sounds ''A'a''

2 is the hamza

5 is kh

6 is Ta'a