The diacritical accent circumflex - ˆ - is best known to most Britons as an element in the French language, which has been one of the most widely studied in British schools for a long time. It is placed directly above the vowel which it marks, as â, ê, î, ô and û. (In French it is always used with a vowel.) The circumflex began as a mark to indicate a long vowel. In French, it serves sometimes to distinguish two words with related vowel, but not identical, vowel sounds, as in cote ('level' or 'mark') and côte ('rib' or 'coast'). For many English-speaking learners of French as a foreign language, the circumflex can give clues as to the meaning of a word. It often marks a contracted form of the word, most often because an '-s-' following the vowel has been lost over the centuries. The modern English word 'hospital' derives from the medieval Latin hospitale meaning 'a place for the reception of guests'. (the word for 'guest' - and 'host' - was hospes in Latin.) In modern French, the word is hôpital. Hospes itself gave English the word 'host' - and in French, it has the form hôte. You will see this in restaurants where you may be offered a table d'hôte menu.
It is not necessary to use circumflexes when writing English. If you are using a word that has become a regular borrowing, like 'fete' (or fête - again with a missing '-s-': the word is the same as English 'feast', and Latin festa lies behind both), you do not need to use the French diacritical accent. But academics like to claim, or pretend, that they are familiar with foreign languages, and so you should do your best to use circumflexes (CORRECTLY) when writing formally, and ther efore in academic writing. Some pairs are 'paste' (English) and pâté (French); 'mast' and mât; 'beast' and bête; 'oyster' and huître; and 'compost' and compôte.
In 1990, the Académie française accepted that the circumflex was no longer needed over '-u-' and '-i-', except to avoid ambiguity. This is not universally agreed. Wikipedia reports that "In February 2016, the Académie française decided to remove the circumflex from about 2000 words, a plan that had been outlined since 1990. However, usage of the circumflex would not be considered incorrect."
In languages other than French, the rules are different. AWE will not go into this.