I disagree partly with this answer claiming that 'aa' and 'ağ' are identical.
The sequence aa does not appear in Turkish words unless they are of Arabic origin, and the proper pronunciation of the aa sequence is not a single lengthened a sound, but rather two separate vowel sounds. Take for example the word cemaat (more properly cema'ât), which in its Arabic form has an 'ayn consonantal sound between the two a sounds. The same goes for the word müracaat, although I concede that the standard pronunciation of the doubled aa in these and similar words is often more akin to ağ or â.
The letter â comes mainly from Arabic loan words where it palatises the preceding consonant and/or lengthens the vowel (but Turkish orthography used to use it for French words as well: plân). Palatisation occurs with the letters k, g and l, but there is sometime ambiguity: kâtil has a long a but an unpalatised k; lâkin has a short a but palatised l.
The sound of ğ after a varies depending on the dialect of Turkish, but in Istanbul Turkish, you'd have to say that ağ and â are identical. The same does not hold true in, say, Eastern Turkey, where you will hear the ğ.