If the residue curve (solid green) is within the region defined by the dotted green lines, that indicates that any information in the residue is statistically insignificant for those lags. The dotted green lines define a 99% confidence region of statistical insignificance of the data.
The residuals are strongly correlated at lag 0 and hence the top plot value at t=0 is always 1. That is not a problem. You probably have a good model since the residues are not correlated with the input. The residuals themselves show some correlation at small lags. This could mean that the residuals are colored. If capturing the nature of the noise (that is, linear filter that is coloring a white noise disturbance) is important, pick a model structure with higher order for the noise component. For example, if you are using an ARX model, you would consider using an ARMAX model while choosing a sufficiently large value for the order of the C polynomial in the equation A(q)y(t) = B(q)u(t) + C(q) e(t).
