Disclaimer: This is meant to be an intuitive explanation without going into mathematical details, given that the original question seemed fairly elementary.
To understand the difference between auto-covariance and auto-correlation, it helps to understand the difference between covariance and correlation.
Covariance is a measure of how much two paired variables v1 and v2 vary in the same way/direction. It is positive when v1 is above its mean at the same time that v2 is above its mean and/or v1 is below its mean at the same time that v2 is below its mean. It is negative when the opposite happens, i.e. whenever v1 is above its mean, v2 is below its mean and vice versa.
It is important to note that covariance only gives you an idea of the direction of the relationship but its hard to interpret the magnitude of the relation, since it is very dependent on the units that are used. (For instance, if your variables are in cm and then you transform it into inches, the absolute value of the covariance will be very different.
Correlation addresses this by scaling the covariance, and putting it into the interval between -1 and 1. Correlation of 1 means your two variables are perfectly positively correlated, -1 means they are perfectly negatively correlated (whenever v1 goes up, v2 goes down), 0 means that there is no correlation at all.
Now, for time-series, the "auto-" prefix indicates that you calculate the covariance and correlation between one variable at time t1 and the the same variable at a later time t1+k. E.g. for k=1, you calculate the covariance and correlation between one time and the next time. The k indicates the difference or lag between the time points, e.g. if you had monthly data, with k=12 you would inspect the relationship between the variables in the first year and the following year.
The auto-correlation coefficients then give you the auto-correlation for each lag k.
Comparing the coefficients for different lags can tell you if there is seasonality in the data - e.g. temperatures tend to change with seasons, so you would expect a higher autocorrelation coefficient at around k=12 for monthly data.