What is Konjunktiv I and when do you use it?

Konjunktiv I is used for reported speech and usually crops up in journalism. For most verbs, the form will differ in the third person singular, which will instead look like the first person singular form. The main irregular Konjunktiv I form is sei (from sein), which conjugates for all persons as well (seist, seien etc.)
Der Präsident hat angekündigt, dass das Problem momentan gelöst werde.The president has announced that the problem is currently being resolved.
Es wird gesagt, die Familie sei vor langem umgezogen.It's said the family moved away long ago.
In both cases, the Konjunktiv I (underlined) is triggered by the preceding verbs of reported speech.