Acetyl chloride will react with an alcohol (with a base catalyst) to give an acetate ester and we get the same product if we use acetic anhydride.
The first step of the reaction is the addition of the nucleophilic alcohol to the electrophilic carbonyl group. The base is important because it removes the proton from the alcohol as it attacks the the central carbonyl group. A common base for this reaction is pyridine. If the electrophile had been an aldehyde or a ketone, we would have got an unstable hemiacetal, which would collapse back to starting materials by eliminating the alcohol. With an acyl chloride, the alkoxide intermediate we get is also unstable. It collapses again by an elimination, this time losing the chloride ion, and forming the ester. Chloride is the leaving group - it leaves with its negative charge. Here is the reaction with the anhydride below:
