Buddhism is often described as a "a guide to life" or "a philosophy" rather than a religion in Western circles. Of course, all religions provide "a guide to life" to some extent, but this is especially true of Eastern religions, including Buddhism, in comparison with Abrahamic religions. It is especially true for the variants of Buddhism prevalent in the West, which often eschew mythological, cosmological and ritualistic aspects and emphasize the practice of meditation and psychological teachings.
Buddhism may or may not believe in souls, gods, or other things, so some Buddhists are Atheist. They believe that suffering is caused by craving or desire, so they practice the ability to ignore greed.
Some Buddhists are heavily invested in supernatural beliefs. Traditions vary, but almost invariably involve belief in karma and reincarnation or rebirth, and the belief in Nirvana (enlightenment) as a means to escape Samsara (the wheel of reincarnation). Buddhists in general hence believe in something that survives death, also an afterlife, or rather next life, that is affected by deeds in this life. This does not necessarily include a god, however there are some "rituals" that are mostly meditation techniques.
Buddhism nevertheless is differentiated from most other religions in its 'empirical' outlook. The Buddha developed techniques for introspection and a body of teachings on what these revealed to him and how to use them to directly perceive the same truths - the core of which is the ability to prevent suffering by severing attachments. The emphasis on direct observation is considerable, and the Buddha often refrained from answering metaphysical questions and instead focused on these more psychological/technical teachings.
At the same time, the Buddha instituted a class of monks, the Sangha, and admonished all to 'put their faith in the Buddha and Sangha', acting against the skeptical teachings. Buddhism also fails to put the fruits of introspection into skeptical tests, and the Buddha believed his own introspection. Buddhism also suffers from popularization, as in e.g. stories about Buddha's supernatural strength as a child or prayers offered to the Buddha.
Buddhism at its heart is therefore founded on psychological truths that probably are largely correct and on productive techniques of observation, but suffers from lack of scientific rigor and a scientific understanding of the way the brain and nature as a whole works. Many atheists, most notably Sam Harris, hope that a more rigorous examination of Buddhist meditation techniques and spirituality in general will lead to a new kind of scientific spirituality, ridding itself of 'Buddhist' aspects and retaining instead universal scientific truths about the world and our psyche.
It should be noted that Buddhism's key insight was shared by the ancient (Greek and Roman) Stoic philosophers, who also taught removal from attachment for precisely the same reasons, and had developed techniques to achieve it - ones based largely on reason and contemplation rather than meditation. Unfortunately, the bulk of the Stoic corpus was lost in the era of Christianity.
In modern times, Positive Psychology is starting to examine happiness and the various techniques to achieve it scientifically, and neuroscience also attempts to understand related issues. It is too early to tell whether the Buddhist psychological tools and meditative introspection techniques will truly prove useful in this endeavour.
Zen differs from many other schools of Buddhism in that it focuses heavy on meditation, enlightenment, and the study of "koans" (parables containing truths that are not directly accessible to the rational mind). It deals very little with reincarnation and supernatural beliefs, and doesn't involve much "moral instruction".