Oxidation states are whole numbers assigned to individual atoms that are part of molecules or ions. They serve to describe oxidation and reduction reactions. An increase in oxidation state is associated with oxidation (loss of electrons) while a decrease is associated with reduction (gain of electrons).
There are multiple sets of rules how to assign oxidation states to atoms (see for example section on oxidation reduction reactions here or what to do for organic molecules here). Sometimes, there is disagreement how exactly to assign oxidation states (it is a conceptual number, you can't measure it), but in all cases, the sum of the oxidation states of all atoms in a molecule or ion or complex has to equal the net charge of that species. This means you have to (artificially) assign each electron to an individual atom based on specific rules or based on electronegativity scales.
To give examples of oxidation states, the carbon in methane has an oxidation state of minus four, while the carbon in carbon dioxide has an oxidation state of plus four. Hydrogen is mostly assigned an oxidation state of plus one, and oxygen mostly an oxidation state of minus two.
When methane is burned (i.e. reacts with elemental oxygen), it is oxidized to form carbon dioxide in an oxidation process. At the same time, the elemental oxygen is reduced to form water and parts of the carbon dioxide. Together, this is called a redox reaction, and there is a transfer of electrons (electrons are available from the oxidation half reaction, and are required for the reduction half reaction).