Confused with oxidation numbers
According to the IUPAC gold book, oxidation state is:


  A measure of the degree of oxidation of an atom in a substance. It is defined as the charge an atom might be imagined to have when electrons are counted according to an agreed-upon set of rules: (l) the oxidation state of a free element (uncombined element) is zero; (2) for a simple (monatomic) ion, the oxidation state is equal to the net charge on the ion;...


Let's consider your reaction $Cu + 2Ag^+ \longrightarrow Cu^{2+} + 2Ag$ The superscripts represent charges. Reaction equations typically don't include oxidation states. In the reactants, $Cu$ has no charge and thus an oxidation state of 0 (rule 1). $Ag^+$ has a charge of +1 and thus an oxidation state of +1 (rule 2). In the products, $Cu^{2+}$ has a charge of +2 and thus an oxidation state of +2 (rule 2). $Ag$ has no charge and thus an oxidation state of 0 (rule 1).

In molecules, oxidation state can be thought of the charge of an atom when all bonds are treated as ionic. For example, in $H_2O$, Both $O-H$ bonds are polar and, since $O$ is more electronegative, all bonding electrons can be considered to belong to $O$. Thus, $O$ has 2 bonding pairs and 2 lone pairs for a total of 8. $O$ has 6 protons so it's oxidation state here is $6-8=-2$. Each $H$ has no electrons and 1 proton for an oxidation state of +1. The total oxidation state of the molecule is $2(+1)+-2=0$, which matches its charge.