cashier's counter; checkout counter
Refers to the physical counter where payment is made, such as in a shop, restaurant, or ticket office.
Please pay at the cashier's counter over there.
There was a long line at the checkout counter.
会計 refers to the act of paying or the bill itself, whereas 勘定場 is the physical place where payment is made.
Compound of 勘定 (かんじょう, 'calculation, bill') and 場 (ば, 'place'). The exact historical derivation is uncertain, but the word transparently means 'place for settling accounts'.