Good without God
It's very difficult to find a nice, short, succinct phrase that neatly sums up what is Humanism, but the catchphrase "good without God" is a pretty good fit. However I'll try to go a bit beyond that, keeping things simple while also accurate.
Religion and morality
Throughout history and in most places, religion and morality have walked hand in hand. Moreover, religion was also involved to a great degree in providing explanations for natural phenomena that people could observe - from volcanoes to rainbows. They may not have been the correct explanation, but they served as a stop-gap measure.
At the same time however there have also been schools of thought that separated these things from religion - various philosophers from the time of the ancient Greeks until today explore morality and ethics with no mention of religion, or examined the world around them as a natural phenomenon, not something that involved magic or miracles.
The inflexibility of religion
One thing that was always quite obvious is that when morality comes from religion, it is very hard to change things. What might have counted as a good, moral principle to live by a century ago could be seen as archaic, barbaric - even immoral - today. This is a problem that plagues religions to this day. As the world came to realise that equality between men and women is a good moral principle, that there's nothing wrong with being gay, or that enforced belief is bad, religions tended to resist these "new" ideas because to do otherwise would go against "God's will". Interestingly, you can even see this within the religions themselves - in Christianity's gospels, Jesus often uses his wits to get around the Jewish religious laws that he already saw as problematic. He did away with stoning the adulteress and permitted certain types of work on the Sabbath - but then of course, once Christianity's own rules got set in stone, it wasn't long before the story repeated itself.
The reason for this is that once a rule has been attributed to a god it becomes difficult for anybody to revise - after all gods are supposed to be infallible. It often takes either a new god, or a new prophet who can convince people that he or she just received an update from God, and he's saying that the old revelation somehow got distorted or something, the new one's better.
The same applies when religion is used to provide an explanation for some natural phenomenon. That is why to this day, the teaching of science is either avoided or undermined in a number of countries around the world - the scripture claims that the universe was created by a god in a particular way, and science has now revealed that things happened in a different way.
Humanism relies on human reason for morality, and on science to learn about the universe. These are more reliable for one particular reason above all - as humans we know we make mistakes, so there is no inordinate resistance to correct something previously held to be true. Moreover, because we recognise this fallibility, we try to take precautions, to verify things properly, check our progress, challenge our positions.
Just as science is always improving itself, correcting past mistakes while learning new things, so Humanism is an ongoing project where we constantly seek to improve ourselves, our society, our values. With no god dictating what is good or what is bad, humanity is recognised as the source of morality (hence Humanism). Or, as Protagoras put it around 450BCE, "Man is the measure of all things".
Humanists not only eschew any kind of divine source of morality, we also avoid any kind of "supernatural" explanation for the world around us. If there's something we don't yet understand we prefer to say "we don't know" than to try to attribute that to some mythical being or mystical force, which is the basis of many modern-day scams.
In the information age, we are constantly bombarded by wild claims, often very well-presented and professional-looking, promising magic snake oil usually as a cure for some real or imaginary illness, or claiming a way to communicate with the dead, or see into the future, . Humanism rejects these modern variations of old scams, relying on science to separate the real deal from the fraud.
The Meaning of Life
One other area where religions often offer an answer is to the question of "Why are we here?", "What is the meaning of life?". Of course their answers are usually "to worship God", which sounds like a rather pathetic reason to exist - created to grovel?
Humanists believe that this life is the only life we have, and that in this life we ourselves, individually, have to set our goals and purpose. We can and do find meaning, beauty and joy in this life we're living. Knowing how a rainbow is formed does not detract from its beauty.
As Humanists there are many things we believe in. We believe in human rights - specifically, we support the Universal Declaration of Human Rights and similar documents. We believe in the freedom of religion - even if we encourage people to embrace reason. We believe in equality - in equal rights for men and women, for people irrespective of race, of sexual orientation, and so on. We believe in ourselves, in the human potential - we do have the capacity to create a better world, to resolve our problems, to overcome our difficulties.
Our About Humanism page contains a number of principles, as well as links to documents that more formally delineate the Worldwide Humanist movement.
What is Humanism? A number of interesting interviews by the British Humanist Association.