Conjugated proteins(protein + nonprotein), nucleoproteins(protein wrapping the nucleic acid ex: histamine), metalloproteins(protein with metal ion), and chromoproteins(protein with pigment molecule).
That makes the 9 protein types.
As to protein structure
primary-AA(amino acid) sequence
tertiary-more folding on top of the secondary
quaternary-two or more amino acid sequences interacting ex: hemoglobin