Data controllers must inform the Information Commissioner of any processing of personal data.
"Notification" is how a data controller provides the Commissioner with the details that form the "register entry" for that data controller. The Commissioner maintains a public register of the register entries.
Most, but not all, data controllers are required to have a register entry. This is only a small part of your obligations under the Data Protection Act - simply having a register entry does not mean that you have fulfilled your obligations under the Act.
All data controllers, whether they are, or are not, required to have a register entry must ensure that all the processing of personal data complies with the remaining provisions of the Act, in particular the Eight Data Protection Principles.