Explain the equivalent bond lengths of the nitrite ion.

Answer 1

To follow up the answer provided by @Lindsay G....

The resonance structures for the nitrate ion are..

O=N-O and O-N=O (with lone pair of electrons on the N atom)

This could be viewed as follows:

O≈N≈O where ≈ represents 1.5 bonds thus both N to bond lengths would be equivalent.

Answer 2

Resonance! Resonance causes bonds to have the same average length.