All beings on this Earth are bound to the 'cycle of suffering'. Suffering does not necessarily mean it must be painful or you must not like it, but it means that they are worldly things. i.e. by being bound to these things, you are 'suffering' as in you are not free and you will not find true peace. If you die, you will be reincarnated to suffer through whatever you deserve depending on your karma in the past life.
Buddhism believes in freeing oneself from the otherwise neverending worldly suffering through detaching oneself from worldly things. Buddhism says you will never cease to feel angry, jealous, happy, or whatever as long as you are attached to worldly things like pleasure, love, and hate. Attachment to physical things also contribute to this suffering cycle. All the things that you see, hear, feel, touch, and taste everyday are all created by your brain, so there is no knowing that they are real and true. Thus, it is fair to say Buddhism is about freeing the mind.
When you meditate, you sit still, close your eyes, and clear your mind. This is so that you are less likely bound by 'outside' things, and you can focus on understanding the facts of life and freeing the mind. Enlightened monks are those who have fully detached themselves from all worldly things and therefore have found true peace. Once they die, they will not reincarnate because they are then free of the cycle of suffering.
Some may say that monks are torturing themselves by denying themselves the pleasures in life. In truth, monks who truly understand Buddhism do not wish for those things because it is 'outside' to them. To them, a new house is the same as a new pair of shoes, all of which is of no importance.
Most Buddhists believe in reincarnation, and strive to achieve Nirvana (enlightenment) and thus get off the 'wheel of rebirth' (in other words, cease being reincarnated). This can be interpreted as "nonexistence," or as graduating to a higher plane of existence -- being reunited with the universe itself.
While generally classified as a religion, Buddhism is probably more accurately described as a spiritual path. However, many Buddhist sects contain strong religious elements.
Buddha himself told his disciples that understanding in Buddhism should be established through personal insight rather than faith or revelation.
Some modern Buddhists, especially in the West, do not accept the ideas of karma and rebirth (after death) at face value.