A method of inference is a way of deriving new knowledge given some initial starting knowledge. The method will specify how this is to be done, and what circumstances must be met for the inference to be valid. There are three major methods of inference: deduction, induction, and abduction.
Deduction involves making logical inferences from premises to conclusions, and is most closely associated with mathematical reasoning. Deductive reasoning begins with some initial premises or axioms which are taken to be true, and certain formal, logical rules of inference that are assumed to be valid. The deductive rules are than applied (often repeatedly) to the initial premises, yielding new propositions which are guaranteed to be true. For example, beginning with the premises ‘Bob is a man’ and ‘if somebody is a man then they are mortal’, then using an accepted deductive rule called monus ponens, it can be derived that ‘Bob is mortal’. This inference, assuming that we do indeed accept the premises and the validity of modus ponens, is proven to be true with certainty.
Induction does not utilise strict logical entailment as deduction does, but rather makes probabilistic inferences based on what is relatively likely to be true given the premises. Often, though not always, induction involves drawing general conclusions from specific cases or observations. For example, if we collect many different samples of a particular species of bird and find them all to have brown features, we may deduce by induction that all birds from this species have brown feathers. Such an inference is not deductively valid, since there are many ways it could turn out to be incorrect (for example we may have just gotten unlucky in finding all brown samples, or perhaps our selected sample was biased in some way, etc). Nevertheless, in general we would regard inferences like this as having a reasonable probability of being true, which is the best that can be achieved using inductive inference. Strong inductive arguments are those in which strong support is provided by the evidence in favour of the conclusion, while weak inductive arguments only provide limited or partial support. This degree of strength differs from deductive arguments, which strictly speaking can only ever be sound or unsound.
Abductive reasoning, also called abductive inference or abduction, is a form of inference which begins with some set of observations or known facts, and infers the truth of some theory or explanation which best accounts for those facts. Abduction is sometimes also described as ‘inference to the best explanation’, meaning that the explanation which best accounts for some phenomena is taken to be the most likely to be correct. Like inductive, abductive inferences can never yield conclusions with certainty, and thus can only justify knowledge to varying degrees of confidence. Though some philosophers consider abduction to be a variant of induction, a notable difference between the two is that inductive inferences need not make any reference to explanation, whereas this is central to the notion of abductive inference. The previous feather colour example of induction, for instance, would not constitute an example of abduction since no appeal is made to formulating an explanation for a body of facts. Abduction is most commonly applied to complex cases in which inferences are made on the basis of large and multifaceted bodies of evidence. For example, the modern theory of plate tectonics is accepted as probably true because it provides the best explanation for a wide body of otherwise inexplicable geologic, biological, and physical observations, not because of any deductive proof of its truth, or even of any simple induction based on past cases or experience.
Inductive and deductive reasoning: brief introductory video from Kahn Academy
Deductive and inductive arguments: useful explanation of the distinction from the Stanford Encyclopedia of Philosophy
Deductive reasoning vs. inductive reasoning: accessible LiveScience article outlining the use of these two types of reasoning in science
Two methods of reasoning: an introduction to inductive and deductive logic
Abduction: Stanford Encyclopedia of Philosophy article discussing and examining the concept