I often see, or am approached with, what I consider to be very erroneous misinterpretations of Buddhism and have thought for a while now of writing my own piece (or maybe eventually pieces, who knows? We'll see how this one goes down) explaining Buddhist beliefs and philosophy. I feel it right to start with a disclaimer that I am, personally, a Tibetan Buddhist, so my own views, practices and beliefs are derived from this tradition and my knowledge of some traditions is sketchy at best - Buddhist traditions are as disparate as Christian denominations - a Tibetan Buddhist temple and its practices are as different from those practised in Theravada Thailand as a Catholic church is from a Gospel church, although obviously both also share some core elements that distinguish them as Christian.
So, let's get started, shall we? I'll address three points here, but if people want to see more written then feel free to ask me questions in the comments, or on Twitter or Tumblr.
Buddhism isn't a religion
This strand of thought often goes along the lines of "Buddhism is more of a philosophy/way of life/etc. than a religion", and there's some truth to this. But Buddhism absolutely is also a religion. Spiritual beliefs are central to almost all traditions of Buddhism, and include beliefs in karma and rebirth/reincarnation. If, in a survey, you asked the question "Does the Dalai Lama have verifiable memories of past lives?" it would very much be the mainstream Buddhist view to agree. While belief in reincarnation or rebirth is, I hold, a spiritual belief rather than a religion-specific one (the vast majority of religions, including early Christianity, have or had a doctrine of rebirth/reincarnation), it seems disingenuous to hold the institutions and teachings which channel these kinds of spiritual beliefs as being something other than religion.
I would totally agree with the statement that Buddhism is not only a religion, but calling Buddhism not a religion at all has always, to me, seemed a bit foolish.
Buddhists don't believe in a soul
While Buddhists typically would never use the term "soul" except when talking to non-Buddhists who might be more familiar with that term, there certainly is an equivalent concept in Buddhist thought. It is variously translated to English as "self"*, "consciousness", "consciousness principle", "spirit", "psyche", or "mind". It's perhaps important to note here that while Buddhists consider the brain important for integrating sensory experience and providing us with an intellect that can form a sensible interpretation of the reality around us (it's also this intellect that's responsible for the greatest illusion of our existence, the illusion that we are our body), the brain itself is not the source of consciousness or mind. The idea of the non-existence of a soul in Buddhism is taken from a misinterpretation of the statement that all that is reincarnated is causes and conditions - that is, karma. But the body is also an aggregate of causes and conditions, a form of karma in itself. If the body forms the causes and conditions for life, then the mind, or soul, forms the causes and conditions for experience, and just as our body is conditioned by our lives (we may have an athletic lifestyle, we may have had injuries that have permanently altered it, we may even have changed it dramatically through surgery and other means), our mind is dependent on mental causes and conditions in the same way the body is dependent on physical causes and conditions, and this, along with the imprints left on it, is what survives after the cessation of the body.
So in summary: Yes, Buddhists do absolutely believe in an equivalent of the soul, but this is not the term preferred by most Buddhists to describe it as it differs from the Judeo-Christian conception.
* A common objection to the concept of a self is that "in the doctrine of Anatta (not-self), the Buddha denied the existence of a self", but this is itself one of the more grievous misinterpretations of Buddhism. The Buddha's doctrine of not-self is a doctrine on where to stop looking for the self, or soul, not the denial of the existence of one. He states that the self, or soul, cannot be found in any of our phenomenal experience of the world, because all phenomena are transitory (impermanent) and selfless, as they are formed from the composite of causes and conditions beyond their own existence.
Therefore, the body cannot be the self. How could it? We inhabit our bodies for the duration of this lifespan, before we are forced to leave the body behind. The word for the body in Tibetan translates to English roughly as "something that's left behind". A large portion of the Buddha's teaching is directed at moving our identity away from our bodies and our current lives to a much broader sense of identity that incorporates the knowledge that while our current lives and bodies are transitory, we are not.
Reincarnation is blind faith
Reincarnation or rebirth is, to people who have had a variety of types of spiritual experiences, not an item of faith but an item of experience. While it is beyond the scope of what I'm writing here to get too bogged down on the evidence for both the existence of verifiable past-life memories and near-death experiences that are held by millions of people around the world, I challenge you to do what a true skeptic would, and do your own research and reading on the topic. But there is certainly enough evidence, both on near death experiences and on past life memories, that a person well-educated on the topic would embarrass the likes of Dawkins who reject the concept out of hand as mere "hallucination", despite the objections of the researchers in these fields themselves, many of whom started out themselves as skeptics.