An acyl group is a general way of referring to an R’-C=O unit that forms the core of compounds like aldehydes, ketones, carboxylic acids, and acyl halides in modern organic chemistry.
This acyl group is easily identifiable with its carbon atom that is double-bonded to oxygen (C=O, carbonyl carbon) with one bond to an R’ group, which can be an alkyl or aryl group.
When R’ of R’-CO is a methyl group (CH3), it is called an acetyl group (CH3CO-), and when the R’ group is phenyl (C6H5), it is called a benzoyl group (C6H5CO-).
As the acyl group accounts for three of the four bonds to the carbon atom, the fourth bond is what distinguishes the functional groups. The ketones, aldehydes, esters, acids, and amides are identifiable due to the variations in the fourth carbon bond.

The acyl group is important to organic chemistry as it is the carbonyl group (C=O) that makes the functional group highly reactive towards nucleophilic addition reactions.
Author's Note: Please note, R’ is simply a way to differentiate an R group. For more details, refer to - What is an R group in organic chemistry?
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Acyl Carbon in Organic Chemistry