I'm surprised that I've never sat down and worked this out before.
My birthday this year is on a Wednesday (today actually) my half Birthday is on a Tuesday. Next year they will both fall on a Thursday since that will be a leap year. I like Thursdays. I was born on a Thursday on a year just after a leap year.
Have you ever tried to follow your birthday's precession around the week? Doing some maths, you quickly realise that:
365 · 7 = 52 remainder 1 52 · 4 = 13
For administration purposes, a month is usually defined as a four-week period.
I don't know about you, but I use the Gregorian Calendar to count my days, but that (only) has 12 months defined. At some point, it was decided to cram an extra month into the rest of our months. They also crammed that extra 1 remainder and, in every four years, another day and a bit for the leapers. The length of a day and the length of a year are time periods that are measurable, imposed upon us by the celestial motions of dust over the course of 5 billion years. Grouping days weeks and months that fit a year is a purely human exercise. [There is an argument (and possibly a blog post) against the anthropic principle here, but I can't be bothered to go through the logic. A lot of artists, poets, philosophers, priests and even (some) mathematicians like to point out the beauty and significance of the world we live in for our needs. As a physicist, I like to poke holes in that particular canvas.]
The important bit that lets one find out the day of a birthday for a given year is that pesky remainder that puts the day of a birthday +1 day later in the week compared to the previous year. The case for a birthday to lie on the same day each year only works with a 5 day week. Of course, there's that extra leap day to cram in that just means that we have to offset the day by +1 every four years (with occasional exceptions).
My birthday is pre-leap day, so the adjustment occurs after every fourth year.
You now probably have enough to figure out the days of my birthdays now.
Thursday, Friday, Saturday, Sunday and when I was four we skipped the Monday and Tuesday was the special day. Followed by a Wednesday, Thursday, Friday, Sunday etc.