Wikipedia has a very good article on the subject. The Dow-Jones Industrial Average was established May 26, 1896, by Charles Dow, editor of the Wall Street Journal. It calculated the average price of a share of stock from the 12 stocks on the list, which was $40.94. The index hit its all-time low of 28.48 during the summer of 1896 - percentage-wise, a far more significant drop than we've had this year.
Now, $40 is still a pretty good average proce for a share of stock, but those 1896 stocks have grown so much in value and have split so many times, that the average is now $10,365.45. Of course, very few companies have shares worth ten grand apiece, but that 10,000-point average once was rooted in reality. On October 9, 2007, the Dow Jones Industrial Average closed at the record level of 14,164.53 - that doesn't seem very much of a reality any more. I wish I had taken my money out last October - but then, what would I have done with it?
It's certainly true that there are other indexes that are far more accurate, but the Dow is the average people are used to. Since Al Gore privatized me out of a steady government pension, most of my income is from investments and I look at the Dow almost every day. The Dow numbers stick in my head, and I can tell at a glance whether my portfolio is going up or down.
So, the Dow is a good tool for telling whether or not I should jump. So far, I haven't jumped.
But the short anwser to Genie's question is that a point is a dollar, but the 10,000-point current average only tells us what an 1896 share of stock would be worth today. If you are 112 years old, this may be highly significant to you. For the rest of us, it's a little hard to root those numbers in reality.

-Joe-