From studies, e.g. with experimental animals, it can be noted that some chemicals cause the development of tumours. As tumours can also be found in non exposed individuals, the number of the tumours should be dose related to conclude that the exposure to the substance is responsible for the tumours. If that is the case, the chemical can be classified as "carcinogenic". It can also be found in such studies, that the substance interact with genetic material, e.g. interaction with reproductive cells, or binding with DNA. That interaction can be confirmed in many in vitro test systems. These chemicals are classified as "genotoxic". As for many genotoxic chemicals tumour formation is also found, these chemicals are classified as "genotoxic carcinogens".
With regard to carcinogenic compounds it should be noted that experts discriminate between "initiators" and "promotors". Initiators are responsible for tumour formation due to an irreversible interaction with DNA, e.g. mutations. This is considered to be the starting point of the carcinogenesis. Promotors can then induce selective enhancement in the iniated cells, leading to the tumours. More detailed information on this topic can be found in the scientific literature and on the internet; see e.g. Britannica: Promotion. Following this hypothesis for carcinogenesis it can be concluded that an exposure to both initiating and promoting chemicals is needed for the development of cancer. It is known that some chemicals are responsible for both initiation and promotion at the same time; these chemicals are classified as "complete carcinogens".
Most experts assume that tumour formation can not start without exposure to an initiating agent. And, that only one molecule of an initiator can already start the cancer process. See for example Encyclopedia.com. Following this model it can be hypothesized that the cancer risk of genotoxic carcinogenic chemicals will follow a linear response, starting at a zero dose. With regard to promotors it means that promoting agents only stimulate tumour formation when the individual was exposed to initiators before. So they postulate that promoting carcinogens demonstrate a treshold mechanism of action. Most experts are however of the opinion that this model is an overly simplified description of carcinogenesis.
There are various organisations that focus on the genotoxic or carcinogenic properties of chemical substances leading to a division in different classes of compounds. The most well known classification is developed by the IARC. This classification use 4 groups (where the second group is divided in Group 2A and group 2B). This class division and other classifications can be found on the Internet, e.g. in Wikipedia. From that article it can be learned that the European Union is using another classification than that of IARC, although in many scientific Opinions (e.g. from EFSA) one can find references to the IARC classification of carcinogens.