What does "To-may-to, to-mah-to" mean?

It refers to different ways of saying tomato. It means it doesn't matter whether you say it with a different accent, it's still the same thing. So the expression means: it doesn't matter.

It refers to the George Gershwin song, "Let's Call The Whole Thing Off", which is a duet between two characters from different social classes, and therefore different accents.

You say eether and I say eyether,

You say neether and I say nyther,

Eether, eyether, neether, nyther,

Let's call the whole thing off!

You like potayto and I like potahto,

You like tomayto and I like tomahto,

Potayto, potahto, tomayto, tomahto!

Let's call the whole thing off!

Nowadays, it's often used when someone feels that the same thing is being referred to using different words.

"I think David Beckham is past his best"

"Well, he's not as young as he was"

"Oh, tomayto, tomahto"

It's a way of dismissing a noted or claimed or supposed difference between two things as trivial.

Attitudinally, it combines an acknowledgement of some difference while simultaneously waving that difference away as not worth bothering with.

And because the pronunciation "to-mah-to" is, at least to ears attuned to US pronunciation, connotatively "upper class," the dismissal gains a subtle heft since to insist on the validity of the "obviously" a trivial difference makes one fussy and highfalutin, a stickler and a snob over petty differences. And that's always a no-no.

There's a class game neatly embedded into the expression, which is there even without benefit of the lyrics from which the phrase was spun-out -- testimony to the verbal genius of Ira Gershwin (though I might have guessed it was Cole Porter, who was also capable of such a feat.)

@Theta Thanks for that great link!

Some corrections about the song (though that is tomato/tomahto re: this discussion)

1. Ira Gershwin wrote the lyrics. George the music.
2. Other lyrics are:

But oh! If we call the whole thing off, Then we must part. And oh! If we ever part, Then that might break my heart!

The last line is:

For we know we need each other so we Better call the calling off off, Let's call the whole thing off.

They want to stay together, so they call off the calling off. Meaning, these reasons are trivial compared to what we mean to each other.

TO-MAY-TO, TO-MAH-TO. Meaning, It's really the same thing. It doesn't matter how you say it or if you use synonyms. Its still the same thing/meanin/word.

I believe it addresses different options. It's about people refusing to agree on something. So the solution is to call things off