Equations are used to describe chemical reactions in terms of their reactants and products. They can be written as either a word equation or by using the correct formulae.
For equations to be chemically correct, they have to be balanced. This requires both sides of the equation to have the same charge and the same number of atoms of each element.
Half equations are a type of ionic equation that represent either an oxidation (loss of electrons) or a reduction (gain of electrons). Like all equations, they need to be balanced. First you balance the atoms, adding water if you're missing oxygen or H+ if you're missing hydrogen. Then you balance the charges by adding electrons.
Ionic equations show which ions interact directly in a chemical reaction and which are spectator ions. All aqueous, ionic compounds are split into their respective ions to form the full ionic equation. Then the spectator ions are cancelled out to create the net ionic equation. Half equations can be combined to create redox equations for a chemical reaction, but the number of electrons in each half equation must be the same. This can be achieved by multiplying the entire half equation.