Do Buddhists believe in God or gods?

In a creator God, no: the early texts say so directly. Devas (deva, plural devā), celestial beings, are numerous in Buddhist cosmology. Even so, they all remain in saṃsāra: although they live extremely long and have supernatural powers beyond human reach, these gods are subject to change and to the law of kamma, like any other beings. They are regarded neither as the sole almighty creators nor as a source of liberation.

• DN 1 describes the appearance of the first Brahmā, the god of the Indian pantheon closest to an “almighty creator”. As a world-period unfolds again, one being is reborn first in an empty palace of Brahmā; others appear there later, and it concludes that it made them itself, while they take its word for it. Brahmā in the Pali Canon, by the way, is not one being but the name of a class of extremely powerful celestial beings (Baka the Brahmā is one of them). The term is quite different from the impersonal brahman of the Upanishads.
• In MN 49 Baka the Brahmā holds his world to be eternal and beyond death, and the Buddha takes that delusion apart in detail.
• In DN 11 a monk tours the heavens asking where the four great elements cease without remainder; the Great Brahmā, in front of his retinue, first recites his titles of creator and lord, then draws the monk aside and admits he does not know.
• AN 3.61 rejects issaranimmānahetu (literally, “the creation of a lord as the cause”, close to the European “divine design”), the explanation of everything a person feels - pleasant, painful or neutral - as the creation of a lord; it also rejects total karmic determinism and pure causelessness.

Devas (the deities of the canon) live long, have great powers and access to almost endless pleasures. But their lifespan ends too: they die and are reborn according to their kamma, sometimes in lower worlds. So although the suttas call rebirth among the devas a good fruit of virtue (MN 41), it is not the end of the path: a heaven is not final nibbāna and no guarantee of liberation (and may even be an obstacle to it). Iti 83 describes the five signs by which other devas recognise that one of them is about to die, and calls a human birth a good destination for the one departing, since it is in the human world that the conditions for practising the Dhamma are right.

Accordingly, the Buddhist refuge formula is refuge in the Buddha, the Dhamma and the Sangha, not in a particular deva. And none of the three is an object of blind faith.

To sum up, Buddhism is non-theistic with respect to a creator God. It does not deny that deities exist, it just does not give them the roles of creator of the whole universe and saviour. Whether or not a single God or Deity exists, the canon states the basic principle unambiguously: there is no point in relying on any god or God in your practice.

“You yourselves must make the effort; the Tathāgatas only point the way” (Dhp 276).

Exceptions

Everything above applies mostly to the Theravāda tradition and the Pali Canon. By contrast, many Mahayana traditions have well-developed theistic practices of worship and of turning to Amitābha, the Buddha of the Pure Land schools, and to Guanyin, the Chinese form of the bodhisattva Avalokiteśvara, usually depicted as female.

Where to read: MN 49 on the Buddha and Baka the Brahmā; DN 11 on the Brahmā with no answer; AN 3.61 on the doctrine of creation by a lord.