Induction usually refers to a form of reasoning that has specific examples as premises and general propositions as conclusions. For example, arguments such as !Swans 1,2,3, …,n are white, hence all swans are white", takes the specific observations of a finite number (n) of swans been white to a general conclusion that all swan are whites. This was taken by many philosopher of science as the basis of scientific method. However, modern definitions of induction state that any form of reasoning where the conclusion isn't necessarily entailed in the premises is a form of inductive reasoning. Therefore, even inferences which depart from general premises to specific conclusion can be inductive, for example "The sun has always risen, so it will also rise tomorrow". On the other hand, we have deductive reasoning: when the conclusions are entailed by the premises. Contrary to deduction, induction can be wrong since the conclusions are always contingent and not necessary.
The Problem of Induction
Since inductive reasoning was establish in its more restricted form to modern days – where more wide definitions were given – there has been a problem with the justification of the validity of induction. Hume argued that the justification for induction could either be a deduction or an induction. Since deductive reasoning only results in necessary conclusions and inductions can fail, the justification for inductive reasoning could not be deductive. But any inductive justification would be circular.
It’s possible to make probabilistic inductive reasoning, such as "95% of humans who ever lived have died; hence I’m going to die". An account for this kind of reasoning is using Bayesian probability, in that case the conclusion is also a probability and induction is taken to be a way of updating your beliefs given evidence (95% of humans been mortals is a evidence of a high probability of you also been mortal). This account is also a more modern view of how the scientific method works.
Mathematical induction is method of mathematical proof where one proves a statement holds for all possible n by showing it holds for the lowest n and then that this statement if preserved by any operation which increases the value of n. For sets with finite members - or infinities members than can be indexed in the natural numbers -, it suffice to show the statement is preserved by the successor operation (If it is true for n, then it is true for n+1). Because the conclusion is necessary given the premises, mathematical induction is taken to be a form of deductive reasoning and it isn't affected by the problem of induction.