There are five simple steps to calculating the weekday for any date in the 21st century.  We'll use the date March 24, 2006 as an example here. 

1.  "Drop the sevens" from the day of the month.
Since we're working with March 24, 2006, the first step is to drop the sevens from the day of the month, i.e. 24.  You can immediately see that 24 = 21+3, or (7x3) + 3.  So you drop the sevens (subtract 21), and you're left with 3.  The 3 is all you have to remember.

2.  Add the month value.
Now you retrieve from memory the digit corresponding to March from the twelve digit number you memorized in the "Get Ready" step.  The digit is "2".  You add that to the "3" you're holding in memory from the paragraph above.  The result is 5.  Hang onto that number in memory; forget the previous numbers.

3.  Adjust for the number of years since 2000.
This step is similar to the above. You just count the total number of years since the year 2000.  In our example of March 24, 2006, the year value is 6.  Add that to the 5 from the last step, and you have 11.  Drop the seven (11 - 7), and you have 4.  Hang onto the 4, and forget everything that came before.

4.  Adjust for leap days since January 1, 2000.
Only slightly more complicated than the above, you have to count the number of leap days that have occurred since January 1, 2000.  The first one was February 29, 2000 - it's easy to forget the year 2000 in the calculation. 

If you were calculating a date in the year 2050, you would say "ok, four goes into 50 twelve times, so there were twelve leap days after 2000, PLUS the leap day that occurred during 2000."  So the answer would be 13.  Drop the seven and you would have 6 leap days. 

Back to our March 24, 2006 example, it's pretty easy to see there are two leap days (2000, 2004) to account for.  Add this to the 4 you're hanging onto from the previous step, and you have 6.

5.  Pronounce the weekday.
Sunday is 1, Monday is 2, etc., with Saturday being 0 or 7.  Since our result in the previous step is 6, you can now confidently tell your audience that March 24, 2006 was a Friday.  Have a friend check the calendar function on his cell phone to confirm.  Now gloat.

Caution
There is one common mistake that's easy to make in counting the number of leap days.  It is when the date you're calculating falls BEFORE February 29 in a given leap year.  In that case, since the leap day for that year has not yet occurred, you do not count that particular year in your leap year count.  For example, if you're calculating the date February 15, 2008, your leap count would still be two (2000, 2004), even though 2008 is a leap year.  (The day of the week for that date is calculated as 2 (Feb) + 15 (date) + 8 (years) + 2 (leaps) = 27; drop the sevens (27 - 21) to get 6; the day is Friday.