Gandalf is indeed a Maiar, and not Human. He is a spirit incarnated; his return to life after the battle with the Balrog was the result of direct intervention by Iluvatar, which is why it's so remarkable as no other character in LotR/Tolkien has done so except Beren Erchamion.
As to there being no 'angels' in Tolkien; that's the role the Ainur fill. Their alternate identities as Valar and Maiar are somewhat problematic when one considers Tolkien's Christianity, as it could be argued that the Valar/Maiar could be interpreted as deities; however, when one considers them as guardian spirits of Arda, then it makes more sense.
The only 'god' in Tolkien is Iluvatar; and he's in a rather odd position as even though he created Arda, apparently he isn't worshipped by its inhabitants. The only deity worshipped to any degree, that we hear of, is Morgoth, and he is of course Tolkien's supreme Satan-analogue; naturally he's going to desire worship. Eru, on the other hand, never demands worship nor service.