The name used can depend on the location from which the goods or services are sold and usage varies between English-speaking countries. Examples:
Department Store: service desk, sales desk; cash desk (a little old fashioned as most people no longer use cash)
Supermarket: checkout (if staffed) and self-service checkout if you scan your own purchases
Other smaller stores: counter - "pay at the counter, please"
In Australia we use all the above but I am sure there are some differences in the USA and the UK which no doubt others will describe.