Hypermiling is the art of driving to use as little fuel as possible. Taken sensibly it can cut a fuel bill by a useful margin; taken to extremes it annoys other drivers and is not worth the risk. Here is the worthwhile middle.
The techniques that work
The biggest gains come from anticipation. Reading the road far ahead lets you lift off early and coast towards a red light or a queue rather than accelerating up to it and braking hard, which throws away the fuel you just spent. Accelerate gently but purposefully, get into the highest sensible gear early, and hold a steady speed rather than surging and slowing. On the motorway, easing off the top of your speed makes a real difference, because the air resistance you fight rises steeply with speed.
Keep the car helping you
Correctly inflated tyres roll more easily, so check them monthly. Strip out weight you are carting around for no reason, and take roof bars and boxes off when they are not needed, since the drag they add is larger than people expect. Keep up with servicing, because a clogged air filter or dragging brake quietly costs fuel.
Myths and things not worth it
Coasting in neutral down hills saves little on a modern injection engine, which already cuts fuel on the overrun when you simply lift off in gear, and it removes engine braking, so it is not worth the safety trade. Drafting close behind lorries saves fuel and is dangerous, so do not. Switching the engine off at every brief stop helps in long waits but is pointless and annoying at a junction.
Do not become a hazard
The one rule that overrides the rest: never let saving fuel make you a nuisance or a danger. Crawling far below the flow of traffic, braking late to coast, or hovering near other vehicles undoes the point. Smooth, anticipatory, steady driving saves fuel and is also simply good driving. The general fuel advice is on the fuel and driving page.