To “call on the carpet” means to severely reprimand or scold the person who is subject to the figurative summons. In many instances, and probably in its original meaning, it referred to an authority figure chastising a subordinate. In modern usage, however, it might be used to describe a peer-to-peer castigation or even a subordinate admonishing a superior.
The actual phrase “call on the carpet” is an idiom with its usage limited primarily to the United States. To describe a similar reprimand in Britain was once referred to as simply "carpeting" someone. This meaning of the word has fallen into disuse and is rarely heard in modern British spoken or written language.
There are a couple of theories about the origins of the phrase. There doesn’t seem to be a consensus on which is correct. The most common explanation of the phrase’s history describes an interaction between a master and servant. The servant’s work would primarily be conducted in uncarpeted rooms, while the master’s quarters and living area would be carpeted. The servant received a literal “call on the carpet” and was summoned to the master’s carpeted area of the home to be reprimanded, giving the idiom its meaning of a scolding.
The other theory is that the carpet referred to a covering on a table at which judges or magistrates would sit to consider various matters. When the phrase “call on the carpet” came into popular use in the 19th century, it was often used to refer to a matter being considered by an authority. Eventually the phrase “on the carpet” to describe something under consideration became “call on the carpet,” and the meaning changed, probably because it was common for the issue being discussed to be misbehavior and the authorities seated at the table might follow their discussion by reprimanding the person before them.
Other theories, though not as commonly accepted as the first two, exist about the origins of the phrase. One says the word “carpet” with the meaning of a reprimand began around English racetracks where underlings stood on a particular piece of carpet to receive scoldings. Another states that in the Victorian era in certain businesses a man was allowed to have carpet in his office only after reaching a certain level of promotion. Thus authorities had carpeted offices and to receive a “call on the carpet” was to be summoned before someone with rank.