Can a divorced couple remarry each other after talaq?

It depends entirely on what type of talaq was given. If the talaq was revocable (raj'i) — which the first and second pronouncements are — the husband can take his wife back directly during her iddah, without a new nikah or new mehr, simply by clearly resuming the marriage in word or action. If the iddah ends without revocation, or the talaq was the third and final one (making it irrevocable, or ba'in), the couple can only remarry through a brand-new nikah contract, with a new mehr and the wife's fresh consent.

There is one further condition once a couple has reached the third and final talaq: they cannot simply remarry each other with a new contract right away — the woman must first have genuinely married someone else, and that second marriage must have ended naturally (through death or divorce), before she and her first husband can remarry (see Q9 on halala). This condition does not apply to a first or second talaq, only after the third, final pronouncement has actually been reached.

Source: Quran 2:228-230 on raj'i (revocable) talaq and remarriage after the third talaq; general Sunni fiqh consensus.