What it must never see
The divider has no idea what is safe. It hands SIG a fifth of whatever is on the terminal: below zero if the input is reversed, above your board's supply if it is too high, and both in turn if it is AC. DC, positive, under your board's limit, and never mains.
What SIG does with it
The band is what a 3.3 V board's pin may see: 0 V up to its supply. The line is what the divider asks SIG to be. Only the first case stays inside.
Reversed
Plus on the GND screw and minus on the + screw asks for a negative voltage on SIG: a fifth of the input, below zero. Every one of these pins has a protection diode to GND, and it stops SIG a few tenths of a volt below zero by conducting. The 30 kΩ resistor keeps that current small. It is still a current the pin is not rated for, and the sketch reads zero the whole time, so nothing tells you it is happening.
Nothing on the board marks plus and minus. The left screw is GND; which screw is which shows how to check it with a multimeter.
Too high
Above your board's limit the reading stops at its maximum and the pin sees more than its supply: 30 V on the terminal is 6 V on SIG, nearly twice what a 3.3 V pin may take and past even an Uno's 5 V. The divider does not saturate or give up. How high you can go has the limit for each board.
AC
An alternating voltage swings above and below zero, fifty or sixty times a second from a mains transformer. On SIG that is a negative half every cycle, and at 12 V AC the peaks are about 17 V, more than a 3.3 V board can take. The block is for DC. Rectify and smooth AC to DC before it comes anywhere near the terminal, and the reading is then of the DC.
Mains
Never. Not through this block, not through any other block with a header on it, not at any ratio. The terminal is rated 300 V, and that is the terminal's rating only: it describes what its plastic stands, not what the block, the wires, the breadboard or you can survive. Mains measurement needs isolated, rated equipment.
When it does not work
Probably not: the 30 kΩ resistor keeps the current into the pin's protection diode small. But a negative voltage on an analog pin is outside every one of these boards' ratings, so do not make a habit of it. Swap the wires, then check a known voltage reads correctly before trusting the pin again.
Not directly. AC swings below zero every half cycle, which puts a negative voltage on SIG, and 12 V AC peaks at about 17 V, above a 3.3 V board's limit. Rectify and smooth it to DC first, and then check the DC is under your board's limit.
None. There are two resistors and nothing else: no fuse, no clamping diode, no reverse-polarity protection. That is not an oversight, it is what makes the reading exactly a fifth. The protection is you, keeping the input inside the limits on this page.
A car battery is about 12 V at rest, which is inside every board's limit. A car's electrical system while the engine runs is another matter: it carries spikes well above its nominal voltage. Measure the battery on the bench, not in a running car.
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