Lay each contract's settlement price against its delivery month and you get the forward curve. It is the market's price for the same barrel at a series of future dates, all quoted today. Reading it is the single most useful skill on a crude desk, because almost every structural trade is a statement about some part of its shape.
The first thing to be clear about is what the curve is not. It is not a forecast. A curve sloping upward does not mean the market expects higher prices, and a curve sloping downward does not mean it expects lower ones. The shape is set by the cost and value of holding the physical commodity over that interval, and those are present-day facts about tanks, freight and interest rates rather than opinions about the future.
There are two names for the two directions. Contango is an upward-sloping curve, where deferred months cost more than prompt ones. Backwardation is the reverse, and it is the market paying a premium for immediate delivery. The mechanism that connects the shape to the physical world is storage and carry: the difference between two months cannot persist far above the cost of storing a barrel between them, because that gap is a riskless trade for anyone with a tank.
Real curves are rarely one clean slope. A curve can be backwardated in the front and in contango further out, which is the normal shape after a supply disruption: prompt barrels are scarce now, and the market does not expect the scarcity to persist. The near part of the curve carries almost all of the information and almost all of the liquidity; the far part is flatter, quieter, and largely determined by the spread market rather than by outright trading.
Because the curve is a set of prices rather than one, it supports a family of derived quantities that a single price cannot. The gap between two months is a spread. The cost of maintaining a position through a roll is roll yield. And the whole curve reduced to a few numbers is what curve factors do.