I was born the 26th of April 1711, old style, at Edinburgh. I was of a good family, both by father and mother: my father's family is a branch of the Earl of Home's, or Hume's; and my ancestors had been proprietors of the estate, which my brother possesses, for several generations. My mother was daughter of Sir David Falconer, President of the College of Justice: the title of Lord Halkerton came by succession to her brother.
Hume doesn't note here that the 'Hume' spelling was entirely his own innovation: none of the other Homes at that time chose to use it. David's reasoning was that it was only if you spelled it that way that the English would pronounce it in anything like the correct way, and David was an Anglophile, and spent much of his career trying to be accepted by respectable English society.
'Old Style' indicates that the date was according to the Julian rather than the Gregorian calendar. Britain only adopted the Gregorian calendar in 1752. So today's the anniversary of Hume's birthday only in a sense; in the Gregorian calendar, unless I miscalculated (which is entirely possible -- I find intercalendar conversions extraordinarily confusing), Hume was born on May 7. But it would still be the anniversary if (as one might and as, in fact, I find that philosophers tend to do) one chose to ignore the fact that it's technically according to a different calendar than the one we actually use.
Incidentally, I think it is also the Julian anniversary of Thomas Reid's birthday; he was born April 26, 1710 -- and I believe that was 'Old Style', too.