Tomato/Potato:
The usual quote is: "You say 'to-may-to', I say 'to-mah-to'", but it's actually a misquote -- it's "You like 'to-may-to', I like 'to-mah-to'" and it's from a song called Let's Call the Whole Thing Off. It's been sung by many different people and the lyrics vary, but here's part of the version sung by Fred Astaire:
You say "ee-ther" and I say "eye-ther",
You say "nee-ther" and I say "ny-ther";
Ee-ther, eye-ther, nee-ther, ny-ther!
Let's call the whole thing off!
You like "po-tay-to" and I like "po-tah-to",
You like "to-may-to" and I like "to-mah-to";
Po-tay-to, po-tah-to, to-may-to, to-mah-to!
Let's call the whole thing off!
It's about differences in dialect, and different pronunciations of the words "either", "neither", "potato" and "tomato" (although I have never heard anyone pronounce "potato" as "po-tah-to"). Because "to-may-to" is the usual American pronunciation and "to-mah-to" the usual British pronunciation, the line is usually quoted in reference to the difference between British and American English.
Old bean:
This is a very old-fashioned term fashionable in the British upper classes a few generations ago. "Bean" simply meant "fellow" or "chap"; the world "old" was simply part of the idiom and was not a reference to the person's real age -- it was almost a sort of mild term of endearment. An "old bean" was a man or (sometimes) a boy; a woman or a girl might be an "old girl"; and a gender-neutral term was "old thing".
All of these phrases are no longer fashionable, but you might find them in books written in the first half of the 20th century (for example in Enid Blyton's Famous Five series, or the works of P.G. Wodehouse).
