I was looking at the conjugation of connaître, and I noticed that there is a circumflex accent in the third person singular, and only in the third person singular. Why is this? The 1st, 2nd, and 3rd-person singulars are all pronounced the same, so why would there be an accent in the 3rd sg.?
Short explanation: Naître used to be spelled naistre, and the third-person singular, naist. But the forms spelled nais retain an "s" in the spelling, so they don't get a circumflex.
Longer explanation: The circumflex on a vowel is often explained as being an indicator of the loss of a following "s" in an earlier form of the word. (Actually, the circumflex is not always an accurate indicator of a "real", pronounced "s" in the etymology of a word; it is also used in the spellings of some words that developed long vowels for other etymological reasons, e.g. chaîne from Latin catena, although this word does seem to have had a spelling chaisne used in the past—another old spelling like that that may be familar to English-speakers is demesne.) Wikipedia gives some more explanation and examples.
Anyway, for this reason, the circumflex doesn't (typically? I don't know if there are any exceptions) appear on a vowel before the letter "s" or "z". (It also doesn't often occur on word-final vowels, as in native words the normal spelling pattern would just use final silent s—e.g. bas is pronounced as if it were spelled *bâ—but I know there are exceptions to this, like the vocative particle "ô".)
A similar pattern shows up in the conjugation of the (literary form) imperfect subjunctive: the endings for -er verbs are -asse, -asses, -ât, -assions, -assiez, -assent. Verbs of other conjugations use other vowels, but maintain the same pattern of ss in most forms vs. a circumflex in the third-person singular.
A pronunciation side note: I am by no means an expert, and this facet of pronunciation varies a lot across times and places (and speakers), but in general a following letter "s" or "z" may have a similar effect on the pronunciation of a vowel as a circumflex accent. Of course, in the varieties of French most often taught to learners today, the circumflex accent mostly doesn't affect pronunciation, so it is not necessary or really useful to know about this. But e.g. in accents with a distinction between something like back /ɑ/ for â and front /a/ for a in most contexts, the same vowel /ɑ/ is used in -ât and -assent. If you're interested in learning about this topic in more detail, here is a blog post: 50 French words using the â sound in Québec — but written without the accent