A functional group is a term used in organic chemistry. They can not exist independently but can combine with different alkyl or aryl groups to produce a special class of compounds called, "Homologous series".
e.g. hydroxyl group is a functional group having chemical formula of: -OH. When it combines with different alkyl/aryl groups it gives a class of compounds known as alcohols. If it combines with some alkyl groups like methyl group(-CH3) or ethyl group(-CH2CH3), it gives methyl alcohol(CH3OH) or ethyl alcohol(CH3CH2OH) respectively. However when it combines with aryl group like phenyl(-C6H5), it gives phenol(C6H5OH).