Compliance is adherence to laws, regulations, standards and procedures. It may refer to obeying laws, following other legally biding requirements (such as those of a regulator), following accepted standards even if they are not legally binding, or even simply complying with an organisations internally determined rules and policies.
Regulation has tended to tighten and become more complex over the years and the cost of ensuring compliance in the financial sector has naturally risen as well. This has been particularly true in the US with regulation such as Sarbanes-Oxley is very expensive to comply with.
Financial institutions" compliance departments that become larger and busier over the years. Despite widespread complaints about costs, few people doubt that regulation is necessary, although the extent of regulation that is desirable is a more contentious issue.
The main source of regulation in the UK is the FSA