Nomenclature of polyatomic ions
Initially the idea was that -ate suffix is reserved for a mononuclear acid with highest oxidation state, while -ite suffix was reserved for a mononuclear acid with second highest oxidation state. The per-...-ate construct was reserved for anion with 'extra' oxygen, typically a peroxocompound and hypo-...-ite construct was reserved for acid extra low on oxygen.

However, since for halogens $\ce{XO4^-}$ anions were obtained rather lately, perhalogenates has this formula and simple halognates are $\ce{XO3^-}$. Mn subgroup (Mn, Tc, Re) inherited this in the sense that highest oxidation state corresponds to per- ... -ate construct 

Similarly, ortho- prefix was reserved for mononuclear acids with highest number of hydroxogroups possible, while meta- prefix was reserved for an acid with one water molecule less (typically, polymer/oligomer) 

Putting it all together using phosphorous as an example


Perphosphate $\ce{PO5^{3-}}$ 
Ortophosphate - $\ce{PO4^{3-}}$
Metaphosphate - $\ce{(PO3^-)_n}$ 
Phosphite $\ce{HPO3^{2-}}$ (since $\ce{H3PO3}$ is actually $\ce{H2[HPO3]}$ for some arcane reasons) Note: depending on the chemistry course you are taking (less advanced courses mainly), some will use $\ce{PO3^{3-}}$ as phosphite
Hypophosphite - $\ce{H2PO2^-}$ (since $\ce{H3PO2}$ is actually $\ce{H[H2PO2]}$


And for chlorine (see above about perhalogenates)


Perchlorate $\ce{ClO4-}$
Chlorate $\ce{ClO3-}$
Chlorite $\ce{ClO2-}$
Hypochlorite $\ce{ClO-}$


Not every degree needs to be present, for example only simple carbonates and percarbonates exist, and sometimes extra stages (usually polynuclear) occur, such as dithionites. This system is widespread in areas heavy on traditions, such as medicine and heavy industry.

There is a gentle pressure from IUPAC to move to much less confusing systematic naming, but the success is limited, especially considering widespread chemicals, which are familiar anyway. It is similar to situation with RJ plugs: everyone recognises phone and Ethernet connectors, but very few care about their full formal name.