In this sentence – You've got to make allowances for their British accents – what does "make allowances" mean?
It means you've got to tolerate their British accents – sounds like some American talking.
When one makes allowances for someone or something, one takes into consideration things that they normally ignore. For instance, one audio version of the Harry Potter books is read by Stephen Fry, a British comedian, actor and writer. He's brilliant but if you're, say, American, you have to make allowances for his British accent – got to get used to a bit of Fry – before you can fully enjoy it.
People make all kinds of allowances every day. If, say, you've got an appointment this evening with a friend at the other part of town, you've got to make allowances for Beijing's notorious rush hour traffic – that is, if it normally takes you 45 minutes to get there, you may want to leave a full hour in advance this time just to be on the safe side.
Mark Twain even made allowances for kings. "All I say is," Twain wrote in Huckleberry Finn, "kings is kings, and you got to make allowances. Take them all around, they're a mighty ornery lot. It's the way they're raised."
And Rudyard Kipling advised us to make allowances for practically everything in his poem If:
If you can keep your head when all about you
Are losing theirs and blaming it on you,
If you can trust yourself when all men doubt you
But make allowance for their doubting too,
If you can wait and not be tired by waiting,
Or being lied about, don't deal in lies,
Or being hated, don't give way to hating,
And yet don't look too good, nor talk too wise:
If you can dream – and not make dreams your master,
If you can think – and not make thoughts your aim;
If you can meet with Triumph and Disaster
And treat those two impostors just the same;
If you can bear to hear the truth you've spoken
Twisted by knaves to make a trap for fools,
Or watch the things you gave your life to, broken,
And stoop and build 'em up with worn-out tools:
If you can make one heap of all your winnings
And risk it all on one turn of pitch-and-toss,
And lose, and start again at your beginnings
And never breathe a word about your loss;
If you can force your heart and nerve and sinew
To serve your turn long after they are gone,
And so hold on when there is nothing in you
Except the Will which says to them: "Hold on!"
If you can talk with crowds and keep your virtue,
Or walk with kings – nor lose the common touch,
If neither foes nor loving friends can hurt you;
If all men count with you, but none too much,
If you can fill the unforgiving minute
With sixty seconds worth of distance run,
Yours is the Earth and everything that's in it,
And – which is more – you'll be a Man, my son!