what is meant by biased and unbiased Autocovariance sequence estimate?
In spectral estimation, one estimates "spectra" and other values. And one is interested in knowing whether the estimate $\hat{x}$ converge to the "true" but unknown value  $x$. It is unknown, for instance because one has only access to a finite quantity $N$ of samples to compute with, a limited number of realization, because ergodicity is not assured, etc. So we want to know whether  $\hat{x}_N\to x$ as $N\to \infty$. The convergence can be thought in different ways, and the most common is the mean square sense: does

$$\lim_{N\to \infty} E(|\hat{x}_N-x|^2)=0$$
where $E$ is the expectation? This can be really complicated to prove in practice. Thus, often people are satisfied when simpler conditions are met. And there are two necessary conditions for the above: 

$$\lim_{N\to \infty} E(\hat{x}_N)=x$$

$$\lim_{N\to \infty} E(|\hat{x}_N|^2)=0$$

The first is the asymptotic unbiasedness, the second the vanishing variance. Together, they define consistent estimators. So, a biased autocorrelation estimate won't converge to the true autocorrelation. And derived estimators (like periodograms) are unlikely to converge as well.