With the noun ring used in the sense of a circular mark or band, the phrase bathtub ring, also ring (a)round the bath(tub), designates a dirty water-level mark left on the inside of a bathtub after it has been drained, caused by a combination of hard water and a build-up of soap scum, oils from bath products, etc.

Several of the texts containing the earliest occurrences that I have found of the phrase bathtub ring, also ring (a)round the bath(tub), are interesting for what they reveal about the relationships between men and women—these texts are as follows, in chronological order:

1-: From the following advertisement, published in the San Francisco Examiner (San Francisco, California, USA) of Thursday 28^th May 1914 [Vol. 100, No. 148, page 9, column 2]:

At YOUR Grocer

It removes the “ring” around the bathtub with a single wiping.
It cleans the toilet bowl and wash basin in short order and makes them antisceptic [sic]. Use it on your hands afterwards and it will not only clean them but leave them so soft and velvety that it will surprise you.
There is no household Cleanser that will do as much.

Handy Andy
A Generous Can 10c
Cleans EVERYTHING
One of the 20-Mule Team Products

2-: From an advertisement for Economy Water Softener, published in the Spokane Daily Chronicle (Spokane, Washington, USA) of Wednesday 14^th April 1920 [Vo. 34, No. 177, page 12, column 6]:

Hard Water Leaves That Gray Ring Around Your Bath Tub
every time you bathe. It’s the union of the minerals in hard water with the fatty acid in the soap that will not dissolve—therefore settles on the bath tub and leaves the ugly ring.

3-: From the column Elizabeth Ann says, published in The Kansas City Kansan (Kansas City, Kansas, USA) of Sunday 26^th June 1921 [Vol. 25, No. 357, page 4 A, column 3]:

A friend of Elizabeth Ann writes to inquire “WHY does the average man think he was born with the special privilege of leaving an inch-thick crusted ring ’round the bath tub, after he has cavorted therein? The traveling man leaves it for the hotel maid. Very well, he is paying for service. But WHY does the unmarried man at home expect the repulsive task to be done by mothers and sisters; the married man by his wife; and the boardin’-house-man by the victim who follows him? I WANT TO KNOW!”
Gosh, m’dear—I don’t know! But he usually DOES! What EXCUSE have YOU to offer, Mister Man? Let’s hear from you!

4-: From the column Flickers, published in the Elmira Star-Gazette (Elmira, New York, USA) of Thursday 26^th January 1922 [Vol. 15, No. 175, page 8, column 7]:

BUT THEY SELDOM DO.

A model
Husband is one
Who always removes
The ring around
The bathtub after
He uses it.

5-: From the column Jeffersonian Siftings, by W. H. Himes, published in the Jeffersonian Democrat (Brookville, Pennsylvania, USA) of Thursday 9^th March 1922 [Vol. 47, No. 10, page 6, column 3]:

No gentleman will leave a ring around the bath tub, but it is considered good business, in order to get ahead of an opponent, to cause rings to form around fathers and mothers eyes, worrying where the price for the next meal is going to come from.

6-: From the column Miami Rambles, published in The Miami Herald (Miami, Florida, USA) of Saturday 30^th December 1922 [Vol. 13, No. 35, page 4 A, column 6]:

“Different climes, different conditions” is probably the only consoling philosophy Rambles can offer to those of our visitors who are amazed, each morning after the daily plunge, to note that there always seems to be a ring around the bath tub.

7-: From The Porter News (Porter, Oklahoma, USA) of Thursday 1^st February 1923 [Vol. 1, No. 36, page 2, column 2]:

Speaking of indoor games how about ring around the bathtub?

8-: From the column Pointed Paragraphs, published in The Roanoke Times (Roanoke, Virginia, USA) of Friday 16^th February 1923 [Vol. 73, No. 40, page 6, column 2]:

Cause For Thankfulness.

One’s troubles are hardly ever so bad that they mightn’t be worse and we often tell our wife, as she comments in her distressed and perhaps somewhat critical way on the ring around the bathtub after we’ve been there, that she ought to be darned thankful she isn’t married to the ashman.—Ohio State Journal.

9-: From the column All Over Ohio, published in The Cincinnati Enquirer (Cincinnati, Ohio, USA) of Wednesday 11^th April 1923 [Vol. 80, No. 101, page 4, column 8]:

Woman should train herself to think in the terms of the larger good, declares the Ohio State Journal, and it is certainly better that a husband should leave a ring around the bathtub than that he should leave it on himself.

10-: From the column Around the Town, published in the Buffalo Evening News (Buffalo, New York, USA) of Wednesday 16^th May 1923 [Vol. 86, No. 31, page 18, column 2]:

IT WAS FAR ENOUGH DOWN on Delaware so that almost the last bit of fallen aristocracy would have held its head in shame to see what it had sunk to. But the landlady, slattern and with pointed yellow teeth sticking out of her jaws, was saying to the timid inquirer for rooms: “Yes, I know exactly what you want. A small room, about $3 or so. But you won’t get it on this street. Yes, you’d better get off this street and try several blocks over. Rooms are pretty expensive on this street.”
And in the bathroom there was a dirty ring around the bathtub; on the backstairs was a heap of trash that could not be accumulated in one day of the life of a rooming house; downstairs, some dirty children are yelling and scrambling over the furniture that cluttered up the front hallway. . . . But it was Delaware avenue, and “the rooms are pretty expensive on this street.”

11-: From a letter to her “Dearest Daddy”, written from New York City by ‘Pam’, in Chasing the Literary Rainbow in Gotham, published in the Sunday Eagle Magazine (Brooklyn, New York, USA) of Sunday 29^th July 1923 [page 2, column 2]:

Situation in the house is about as was, only more so. […] On account of frequent suspicious noises without, during my morning tub, I have ceiling-waxed the bathroom keyhole, and have also composed and erected a sign requesting slovenly members to kindly remove their personal bathtub rings after using, but presume it should be written in at least five languages. Time will tell.