Induction is the process of drawing an inferential conclusion from observations - usually of the form that all the observed members of a class defined by having property A have property B. The classic example is that of determining that since all swans one has observed are white that therefore, all swans are white. In inductive reasoning, one is extracting a general principle from specific observations.
Induction can be compared with deduction which works the other way around: one might observe that all men are mortal, and have good reason to believe that Socrates is a man, and can therefore deduce that Socrates is mortal. This deduction cannot be overturned by further observation in the same way that an inductive conclusion can be: one might have reason to doubt that all men are mortal, or one might find out that Socrates is actually an angel. But if both the two premises hold, the deductive conclusion follows.
Inductive reasoning is an important feature of all forms of human reasoning and good epistemic practice, and is seen by many as the foundation of empirical science. Most empirical sciences rely on reasoning from specific observations of the natural world to general laws and theories.
Philosophers from antiquity have questioned whether or not induction leads to knowledge. David Hume argued in An Enquiry concerning Human Understanding that induction needs a justification. What justification can there be for induction? Hume doubts that a deductive justification can be offered for inductive reasoning. Instead, the reason why we use inductive reasoning is because it seems to work - ask any scientist or anyone with the basic levels of common sense required to be a good knower and they will usually say that they know inductive reasoning works because of successful examples of inductive reasoning. This just seems to beg the question.
A more contemporary issue with induction was suggested by the American philosopher Nelson Goodman in his 1966 paper "The New Problem of Induction" (and subsequently in Fact, Fiction, and Forecast. Goodman illustrates the problem with reference to a predicate "grue". We inductively believe that all emeralds are green. But to say something is "grue" is to say that it is green if observed before a certain time and blue if observed after that time. For any particular emerald, if we are satisfied in applying "is green" as a predicate to it, we should be equally satisfied to apply "is grue". Why, then, in our inductive reasoning do we reason from the greenness of every emerald we have seen to the greenness of all emeralds and not the grueness of all emeralds? Both are possible conclusions from the same data, but we seem to not bother to infer grueness.