An overview. The word 'induction' is used in two quite different ways in
logic. There is traditional logic and there is modern
mathematical logic. The mathematical one doesn't replace the traditional one, which remains correct, but the two are used in different spheres.
More people these days are more familiar with mathematics than with traditional logic so this might need to be stated clearly for those who are confused.
Traditionally, arguments are either deductive or inductive. Deduction gives certainty, induction only gives likelihood. If you can prove deductively that something is true, there is no way it can fail to be. If you argue inductively towards a conclusion, you only demonstrate a high probability that the matter will turn out be to true. (A third possibility, formulated in the late nineteenth century, is abduction.)
Examples: the sun has risen every day so far, all swans so far examined are white, and all known crows are black. So we can presume, by induction, that the sun will rise tomorrow, that all swans are white, and that all crows are black. David Hume articulated the classic argument that we can never entirely rely on such inductive reasonings, no matter how well grounded they seem to be on evidence. Frivolous thought: Perhaps he was writing just after the discovery of black swans in New Holland.
In mathematics there is a different principle called 'induction' or 'mathematical induction', which is deductive. Mathematical induction is a valid deductive argument and gives the answer with certainty (if the conditions are fulfilled): it is not inductive in the traditional sense and does not suffer from Hume's Problem of Induction.
The usual form of mathematical induction works by proving a formula for (i) some trivial case such as k = 0 or k = 1, and (ii) a general conditional that if it's true for arbitrary k then it's true for k + 1. If these jointly hold then it's true for all natural numbers k.
A superficially stronger form is called strong induction. This requires for condition (ii) that the formula holds not just for the one k previous to the one we're now considering, but for all k less than it. Usually this is harder to establish but with some problems it's easier. The end result is the same, that it's true for all k.
There is a technique called transfinite induction that uses the same kind of method, but instead of being limited to all natural numbers, it is applicable to all ordinals at every level of infinity. This has three branches: (i) a trivial initial condition such as k = 0, (ii) a successor condition that if it's true for an ordinal k then it's true for its successor ordinal k + 1, and (iii) a limit condition that if it's true for all k less than some limit ordinal λ then it's true for λ.
Transfinite induction is not however inherently true, in the sense that it's not in the usual axioms of set theory: it is equivalent to the Axiom of Choice, to the Well Ordering Theorem, and to Zorn's Lemma. It is therefore possible to have alternative mathematical logics in which it doesn't hold.