Do Buddhism and Atheism contradict each other?

One of the primary tenets in Buddhism, across nearly all forms of Buddhism is part of this list:

Whether Theravada or Mahayana, we do not believe that this world is created and ruled by a god at his will.

So, strictly speaking, one may believe and practice a form of Buddhism, and also believe and practice a form of Atheism without conflict.

The other answers have covered this from the Buddhist side, I'll try to answer from the Atheist side.

"Atheism" in general is not a belief, it is a lack of belief. It's a convenient label for an absence that is sometimes treated as a 'thing'. In this respect it is like "vacuum" (absence of matter), "dark" (absence of light), or "freedom" (absence of restriction). "Atheism" is the absence of theism (Belief in one or more gods). Atheists include people who have some idea of what a god is and believe none exist, people who think the word "god" is meaningless, and people who are unsure if gods exist and any other set of beliefs that does not include "at least one god exists".

So you'll pretty much have to decide if any of the entities claimed to exist in Buddhism (or in any particular variant thereof) are "gods". I'm in the "god" is a nonsense word camp myself ("theological noncognitism" if you really want the formal term for it) so I think the whole question is pointless. You may disagree.

"Atheist" is commonly conflated with other terms like Freethinker/Skeptic, Irreligious, and Agnostic. In case that's what you are after, I'll address them.

Agnosticism is about knowledge rather than belief, and it can be applied to things other than the existence of gods. Knowledge in this sense is belief justified by evidence and reason. Belief without such justification is faith. Of course whether something thought to be justification really is makes things complicated in a way that's way beyond the scope of this answer. It's entirely possible to believe in a god while recognizing that that belief lacks justification. Many Christians even claim faith to be superior to knowledge. My understanding is that Buddhists generally see faith as less desirable than knowledge, but an acceptable alternative where knowledge is unavailable.

Freethinkers and Skeptics are concerned with making our beliefs as close to reality as we can, no matter what that turns out to be. We want to verify those beliefs, and try to keep reverifying in case we made a mistake. We seek knowledge and reject faith and dogma. Ideally, we prefer to admit ignorance over clinging to a belief that lacks proper justification. Although this is distinct from "atheism" it is the group most likely to self identify as "atheist". Buddhism makes unjustifed claims (Existence of a soul, reincarnation of those souls, etc) and so is not compatible with freethought. Were all the claims to be justified by sufficient evidence, this would change. (Ideally, all free thinkers would then accept the claims of fact made by Buddhism. We might still reject the ethical claims.)

Irreligious just means "not religious" and the same way atheist means "not theist". Buddhism is a religion and so, obviously you can't be Buddhist and irreligious at the same time. There is sometimes argument over exactly what constitutes a religion, and Buddhism is probably more amenable to this sort of semantic wiggling than many other religions (see the 'Secular Buddhism' described in some other answers)

Naturalism (In the context of Metaphysics and Epistemology) is the philosophy that the world behaves in a way that we can understand in terms of rules which we can then use to make predictions. Where Freethought is the position that we should seek knowledge, naturalism is the belief that we have a chance (in principle) of succeeding. That which can not be understood this way is "supernatural". Many of Buddhism's claims seem to be supernatural and so incompatible with naturalism. Claims about the supernatural also mean that sufficient justification for those claims is never possible, so this would make it impossible to ever be compatible with Freethought.

One can be an atheist and a Buddhist at the same time, if being atheist just means rejecting the concept of an almighty, all compassionate, creator God. But if you mean rejecting swarga(higher realms), niraya(lower realms) and Samsaric existences, you can't really be called a Buddhist as you are rejecting the teachings of the Buddha. But you can still practice Buddhism at a certain level.

There is the Secular Buddhist movement that would be very compatible with atheistic beliefs. Secular Buddhists tend to reject any supernatural beliefs within Buddhism in favour of pragmatic or evidence based solutions. A good (short) text to look at if you are interested is Buddhism without Beliefs by Stephen Batchelor.

Of course you could say 'Aha but these aren't real Buddhists'. But then you could take that attitude to any group of Buddhists that you find a bit unpalatable and to my mind this is getting towards the No True Scotsman fallacy. In short, if you identify as an atheist then it is perfectly possible to practice as a (secular) Buddhist and be part of a community identifies as both atheists and Buddhists and feel the same way as you do.

I'll take an almost certainly unpopular tack here, and say that atheism enhances Buddhism. What I mean by that, is that atheists tend to reject authority in favor of personal experience and investigation, and I think this goes hand-in-hand with the Buddha's teaching. I believe Buddha would be pleased if we critically examined the esoteric "supernatural" aspects taught by some Buddhists schools, and to determine the veracity of them for ourselves, through practice. I actually am of the opinion that the tenets of Buddhism, including kamma, punabbhava, and even Buddhist cosmology, fit into an atheistic viewpoint. To me, "atheism" is a particular way of looking at the mechanisms by which various things occur.

I remind you, this is an unpopular viewpoint.

Within the canonical texts of the various schools of Buddhism there are various beings that roughly correspond to that of deities, in particular Devas and Brahmas. Both Devas and Brahmas are particular types of beings who have taken very high rebirths as a result of their good Karma. They have very long lifespans and many wonderful benefits.

However they aren't the objects of reverence that they are in other religions. Although one might pray to or request one of these beings for help in worldly matters it is understood that these beings are rather limited, and they eventually die like any other being.

In practical terms rejecting the existence of such beings doesn't really present major problems as it isn't an essential doctrinal teaching. One can be a practicing Buddhist without believing in them.