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", take the specific observations of a finite number (n) of swans been white to a general conclusion that all swan are whites.
Modern views 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 proceed from general premises to specific conclusions can be inductive, for example "The sun has always risen, so it will also rise tomorrow". In contrast, in deductive reasoning the conclusions are logically entailed by the premises. Contrary to deduction, induction can be wrong since the conclusions depend on the way the world actually is, not merely on the logical structure of the argument.
There has historically 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 circular1.
It’s possible to engage in probabilistic inductive reasoning, such as "95% of humans who ever lived have died; hence I’m going to die". This kind of reasoning employs Bayesian probability, in which case the conclusion is also a probability and induction is taken to be a way of updating your beliefs given evidence (finding out that most humans who have ever lived have died increases your probability that you will die).
Solomonoff induction is a formalization of the problem of induction which has been claimed to solve the problem of induction. It starts with all possible hypotheses (sequences) as represented by computer programs (that generate those sequences), weighted by their simplicity. It then proceeds to discard any hypotheses which are inconsistent with the data, and to update the probabilities of the remaining hypotheses.
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 suffices 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.