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Someone who would like to talk to a journalist to provide information without being identified can ask to speak to the journalist "off the record." If the journalist agrees that the source is off the record, the journalist would use the information without naming the source. This is common where the source would face consequences for being quoted -- e.g., the source wants to give information about their employer without getting fired. This is an agreement between the source and the journalist, which means that the journalist must agree to anonymity up front and the source cannot insist on it afterward (e.g. because they realize they said something that can get them in trouble). Journalists generally follow these agreements because it would be harmful to their reputation and ability to get sources in the future if they were known to burn their sources, but there's likely no other recourse for a source that believe that they were "off the record" and where then cited by the journalist.

There are various types of "off the record" that are understood to exist, and people, especially in politics or national security would generally be familiar with them. The most common are "background" and "deep background," with "background" indicating that the source could only be cited with consent (i.e. the journalist can't say "My sources in the White House tell me..." unless the source agrees) and "deep background" meaning that the journalist can use the information but they can't even anonymously cite the source (i.e. no "My anonymous source at the Pentagon tells me...").

If someone who is being interviewed wants to help the journalist uncover the truth about a subject but does not want to risk their job or reputation they might tell the journalist something in confidence. They do this by telling the journalist to keep this off their records of what was said in the interview. The journalist would have to abandon their own morals and integrity to tell someone where they got the information from, which might cause them to lose their job. And even then the source will probably deny that they said it. So it is hard for the journalist to print the information as they are not able to verify to others where they got the information from. So it is not as valuable as if the source was willing to go on the record. However it can help the journalist ask the right question and dig in the right places to uncover the truth through other means.

It's founded on trust. A journalist takes care to report with integrity. So when they receive important information from someone who requests anonymity, a good journalist works to corroborate this information with other facts. Background will only speak to a journalist who they trust will honour their desire for anonymity. Good editors will only publish a story that cites background sources if they trust that the journalist did the work to verify that the facts are at least plausible.

The idea is that if the journalist violated the "off the record" request from the source, then people who could be used as sources would stop talking to that journalist. Obviously, sometimes there may be a reason for the journalist to violate it anyway.

Off the record means the person is asking not to write that information in an article.

It's a matter of trust that the journalist won't compromise a source by revealing too much information.