What are the orbitals and the hybridization of the #["O"_2"NO"]^"- "# ion?
I think you are asking about the hybridization and the orbitals involved in [O₁—O₂—N—O]⁻.
You first have to draw the Lewis structure. There are three possibilities (sorry, they are difficult to draw in this editor):
The third structure is the most stable, but the actual structure is a resonance hybrid of them all.
All the atoms have a double bond in at least one of the contributors. That means they must be sp² hybridized with the left-over p orbitals participating in the π system.
O₁ and O₂ are each sp² hybridized. The O₁—O₂ σ bond involves overlap of sp² orbitals from O₁ and O₂.
O₂ and N are each sp² hybridized. The O₂—N σ bond involves overlap of sp² orbitals from O₂ and N.
The p orbitals from O₁, O₂, N, and O all overlap to form an extended π orbital system over the whole ion.