Two forms of the same thing
On most sets "the advance" means the block at the bottom or back of the regular call sheet headed "Advance schedule." Some productions also issue a standalone advance call sheet, formatted like a real call sheet, a few days before a big day: a night exterior with 80 background, a stunt day, or the first day at a new location.
Either way it's built from the same source as the call sheet, the current shooting schedule, filtered to the days coming up.
What a sample advance block looks like
ADVANCE SCHEDULE: DAY 7, WEDNESDAY MARCH 11 (tentative)
Location: Hollis Farmhouse, 22 Ridge Rd (company move from diner)
Sc. 40, 41 EXT. FARMHOUSE PORCH - DUSK 1, 3 1 6/8
Sc. 44 INT. FARMHOUSE KITCHEN - N 1, 3, 5 2 1/8
Special: rain towers sc. 40; 1987 Volvo picture car; child actor (5), studio teacher
Tentative crew call: 12:00 PM
That short block tells the transportation coordinator to plan the move and the picture car, the special effects crew to get rain towers and water trucks confirmed, and the production office to book a studio teacher. It also warns everybody that tomorrow is a late call, so tonight's wrap time affects turnaround less than usual.
When people read the advance as final
The biggest one is treating the advance as final. A cast member who reads "Day 7: H" in the advance and books a flight will be unavailable when the 1st AD swaps days after losing two hours to rain. Productions protect themselves by stamping "tentative" or "subject to change" on every advance.
The opposite mistake happens in the office. An advance that isn't updated after a schedule change keeps circulating in people's inboxes and gets printed as if it were current. Put a revision date or color on every version, the same way script pages carry revision colors.
How it fits with other schedules
The one-line schedule covers the whole shoot in condensed form, the advance covers the next few days in more detail, and the call sheet covers tomorrow with firm times. Departments read them at those three distances. For a worked explanation see advance call sheets, and the call sheet template includes an advance section you can fill in.