I recently placed a bet on a football parley card in Las Vegas of $50.
The odds were quoted at 8 for 1. Upon winning, I was paid $350 for the
win,(7 to 1) and $50 more for my original bet, a total of $400. I , of
course was expecting to collect $450, $400 (8 to 1) + $50(original
bet). I am a medical doctor, took statistics in college and never
heard of 8 for 1 odds. Please explain.

You got paid 8 times your original bet, hence 8 to 1.  Why do you
think you get your bet amount back on top of that? If you buy a $1
lottery ticket and win $10,000 prize, do you think they give you
$10,000 or $10,001? You don't get your original bet amount back on top
of your winnings.

Since you mention your strong math aptitude, I am wondering how you
figure that 8 X 50 = 450?  I assure you that 8 X 50 = 400.  You can
confirm on a calculator if necessary.

Sory, but you obviously have never been to Vegas. If you bet $100 on a
 straight up 1 to 1 bet, you win $100 and you get your $100 original
bet back, you get paid $200. If it was a 2 to 1 bet you win $200 and
get paid $300. I thought my bet was 8 to 1, so I should win $400 and
get my $50 back-total-$450. Again, the bet was advertised 8 for 1, not
8 to 1. What is the difference.

There is a difference between 8 for 1 and 8 to 1.  In Craps for
example be careful, there are 5 for 1 bets for example.  This means
that you get 5 total for your 1.  5 to 1 means that they give you 5
times your bet.