Just what does the price mean?
The price (big number on the front page, as well as the data points for the graphs) represent the price per barrel of Nymex crude oil futures. This seems to be the most-often reported number when talking about the price of a barrel of oil.
Why does the text show a price increase or decrease while the daily graph seems to show the opposite?
The answer is that the graph only shows movement of the price beginning at about 8:00 AM Eastern time. However, after-hours trading moves the price, and the text reflects the current price relative to the previous day's close, not the current day's open.
Why does the graph sometimes show a big jump from one day to the next?
This is another case of after-hours trading affecting the display (discussed here). The last point on the graph for a particular day shows the approximate closing price at around 4:00 PM Eastern while the first point for the next day is the opening price for that day. In the intervening time, after-hours trading may drive the price up or down.