I think it's first important to note for our ESL/EFL friends that one would practically never have a need to refer to one of these. (I personally don't ever recall hearing any name for this.) If one wants to ask for a bill in a restaurant, the answer is here: how should I ask for a bill in a restaurant politely.
This is the leather (or faux leather) folder that waiters/waitresses use to deliver the check/bill to the customer, who reviews the bill and encloses cash or a credit card back to the waiter. The following are terms used by several online restaurant supply distributors:1
- Guest Check Presenter, Guest Check Holder
- Check Presenter, Check Holder
- Bill Presenter, Bill Holder
- Check Presentation Cover
- Bill Presentation Folder
Most people have never considered what this is called, but would likely use the following:
- (bill | check) + (holder | folder) 2
- So for example, if it was misplaced and separated from the bill, one might ask, "Where's the (black) bill holder?".
- If it was misplaced with the bill, one may simply ask "Where's the bill?"
- Another functional way people talk about something when their mind isn't focused is to describe it in terms of how it is used: "that black thing/thingy that the waiter brings with the bill in it."
Here are a few Google Results:
This is similar to they way I sometimes describe the TV remote control to my wife late at night: "Darling, do you know where the channel changer is?" (Evidently I'm not alone on this with Google
"channel+changer" "remote+control" at 14K results.)
1. www.bigtray.com, www.usaequipmentdirect.com, www.centralrestaurant.com, www.zesco.com, www.foodservicewarehouse.com
2. AmE uses check or bill. BrE uses bill.