When the range of the residuals - as you saw it, or to use another measure of spread, the variance of residuals - changes over time, this is a symptom that goes by the name of heteroskedasticity, from the Greek root words of "hetero" for different and "skedasis" for dispersion or spread. In a cross section, this would be the end of the diagnosis.
In a time series context, however, this is only a symptom of a potentially deeper, underlying condition - lack of covariance stationarity (alternatively expressed - lack of weak stationarity). This is evidenced by the fact that the autocovariance function in your case is a function of time. However, this conclusion isn't completely foolproof because stationarity or lack of it is a condition of the underlying stochastic process and not of the single realization you observed (i.e. the sample that you have collected). In other words, the stochastic process that has produced this series may be stationary but you may have had the bad luck of observing/collecting a very unrepresentative realization/sample. But chances are that if your realization is not stationary then the underlying stochastic process is not stationary.
Lack of (weak) stationarity isn't a death sentence for your series, by any means, but it is a condition that will require some special attention. The presence/lack of stationarity is very informative as to your potential next steps in modelling this series.
Read up on stationarity, transformations of non-stationary series into stationary (concept of "integrated" series), ARMA and ARIMA modelling. This site can guide you well in this journey... just do some searches for some of those keyterms.
