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“”The great tragedy of Science — the slaying of a beautiful hypothesis by an ugly fact.
A hypothesis is a proposed explanation for an event or observation. Hypotheses are testable statements, and are a core part of the scientific method.
"Hypothesis" (like the word theory) has a few different meanings - the common usage, a more exact scientific one and also a meaning in statistics, such as "null hypothesis". The meanings behind "hypothesis" are rarely confused, except when people try to abuse language for the furtherance of ideology, as in the way creationists use the words "theory" and "hypothesis" to try to delegitimize scientific ideas they do not like. Conflating these is often one of the sources of confusion and criticism of science.
In everyday language, a hypothesis means a possible explanation for a phenomenon. This can be an idea that is merely PIDOOMA in nature, or something that follows common sense. Often, it's considered a somewhat "weaker" form of the term theory, as if a theory is only a hypothesis that has more support - but this isn't the case, and the distinction between hypothesis and theory is part of the scientific definitions of the two terms.
In science, the meaning is more precise. As part of the scientific method, a hypothesis is the stage where a testable statement is generated. In this case, a hypothesis is usually an original idea generated from data, or from existing theories, which is tested for its ability to explain and predict phenomena. A successful hypothesis lends support to the theory that produced it, and the successful theory will produce further hypotheses to be tested. For instance, a theory about gravitation would produce a hypothesis about the orbits of planets - the orbit of Mercury being a specific hypothesis that differentiated classical Newtonian mechanics with general relativity as a model for gravity.
There are often a multiplicity of hypotheses on any subject which are slowly (or quickly!) winnowed down to one or two similarly viable possibilities. The ability of an hypothesis to make viable predictions is used to test its validity. The inability of one to do so tends to make it evident that it is not actually scientific.
 General requirements
While it could be said that hypotheses have no requirements in order to actually be considered hypotheses, if they are to progress through to become a theory, they must fit certain criteria. Perhaps most importantly it should be theoretically possible to disprove them — that is not to say that they will be disproved, only that it must be possible. Obviously if they are actually disproved they will not progress.
A hypothesis often gathers together or builds on earlier theories or hypotheses. This was, for example, the case with Darwin's Theory of Evolution by Natural Selection. Although this is a normal part of the scientific process, some fundamentalists seem to think it devalues his contributions.
Often, hypotheses can be tested — unless, of course, they are in the form of the argument from God; but in that case they are not science anyway. However, hypotheses about what happened a long time ago, at the point of the Big Bang for example, are much more difficult to test directly.