Reading a bit of its history it seems like it was a pretty important regional power. What was this power based on? I'm also interested in any reading suggestions about the culture of Ghana in the time period, if anyone has any good ones.
I'm going to assume you mean the Ghana Empire that was located in interior West Africa, more or less at the border of present-day Mali and Mauritania between about 600 CE and 1400 CE.
There's been some interesting new arguments made about Ghana in Michael Gomez' excellent recent book African Dominion. Gomez has set out to argue for the world-historical significance of states that formed in Senegambia and the upper and middle Niger River region between long drying of the Sahara and the beginnings of the Atlantic trade--combined with new work by Toby Green and Francois-Xavier Fauvelle, a genuinely new synthesis account of West African history in the centuries prior to the Atlantic trade is coalescing.
Gomez notes the previous understanding of Ghana as one of the earliest of a serial progression of West African empires--Ghana, Mali, Songhay--each of the more centralized, elaborated and larger than the last. Instead, Gomez argues, the 'cores' of these states--and others like Kanem-Bornu in what is now Chad and the states of Senegambia and Upper Guinea--were simultaneous with one another and essentially influenced by each other. So even when Ghana was a tributary of Mali, it still retained a good deal of its distinctiveness and coherence in administrative, political and cultural terms.
Gomez also observes, however, that we don't have a good account at present--and his only scratches the surface--of the ideological, cultural and social distinctions between these states. It's fairly clear that they had different orientations toward Islam, but perhaps also towards gender, towards social organization or hierarchy, and towards variant ideas about political legitimacy.
The key takeaway here though if you follow Gomez (and on this I think he is unquestionably correct) is that Ghana's relations with other Upper Niger, Senegambia and North African "cores" was pragmatic--that core areas paid off others that were in ascension but that they expected to retain practical authority in their central areas of cultural and social coherence.