You never know when you might want to significantly alter the orbital path of the Earth. Maybe the Sun is going Red Giant and you miss the days when lead didn't melt in direct sunlight. Maybe Earth is about to hit an asteroid. Maybe it isn't, but you want it to. Maybe you want to destroy it: a significant number of methods for destroying the Earth involve moving it by some substantial amount.
Well, it isn't easy. In fact, it's very very difficult. The Earth is very big, moving very fast, and therefore very difficult to stop or even slow down.
Ways of moving the Earth
Ordered approximately by plausibility.
Electromagnetic influence. Traditionally the Earth is thought of as "ground", "neutral". This is because overall it carries almost exactly zero overall charge. But what if it didn't? If there was some way to electrically charge the Earth, by dumping lots of identically charged particles onto the Earth or just ionizing particles already on Earth - a large amber rod might perhaps be in order - then we could use magnetic fields to drive the planet in the direction we wanted it to go. Maybe. Or better yet: the Earth already has a standing magnetic field; perhaps we could construct a cylinder of cable around it, and pass current to move it using Lorentz forces.
I know what you're thinking. Yeah, this is ridiculously implausible. Moving on.
Direct rocket propulsion. Build gigantic, possibly nuclear upward-pointing rocket furnaces, maybe one, maybe four, maybe a million, whatever you can budget for. "Gigantic" as in the size of, say, Belgium. Design them carefully so that when used the rocket engines do not actually just propel themselves through the ground and into Earth where they become useless - you may need to periodically dig them out again after several thousand years' continued thrusting, or else just build new ones over the top.
The most obvious major drawback with this method is that right now there aren't even theories as to how you could possibly build rocket engines of the sort proposed here.
Another, more sophisticated, problem is that the Earth is constantly spinning. You could build an engine at either pole and this wouldn't have any effect, but anywhere else and the constantly changing angle of thrust will cause the Earth to behave somewhat like a loose Catherine Wheel-type firework. Plotting acceleration vectors towards whatever your target is in this case may prove to be a nontrivial problem, solvable only with high-tech computer simulations. Alternatively, as the Earth's angular kinetic energy is negligible compared to its orbital kinetic energy, you might consider diverting a relatively small amount of resources to simply stopping the Earth from spinning at all, before beginning the main project.
Atmospheric considerations are ignored here since it is far more energy-efficient to manually remove the Earth's atmosphere, move the planet, and reinstall it.
Direct matter propulsion. Same method as above, just using gigantic mass drivers/railguns to fire huge quantities of matter away from Earth, instead of a rocket exhaust. The principle here is much the same, with the railguns behaving somewhat like discretized versions of thrusters, providing instantaneous changes in velocity as opposed to sustained steady change. Drawbacks: as above, the momentum change you get is minuscule because you have to subtract off the 11km/s needed to launch the material upwards forever at all. Highly inefficient.
Disassemble, move the bits and pieces, reassemble. The major problem here is figuring out how to pick up big pieces of continental plate without breaking them. It all depends how fussy you are about how the Earth looks afterwards, of course.
Solar sail method. I can't honestly add much to that article except to say that to move the Earth substantially, the sail used is going to have to be pretty big. Like, suspended-from-space-elevators big. Difficult... Colin McInnes, however, suggests an alternative. Construct a huge solar sail with a significant mass. Plan it right, and you can couple it together with the Earth with gravity alone, using the solar wind to balance out the Earth's gravitational attraction. Wait long enough, and the solar wind blowing the sail outwards will take the Earth outward too, since the two are gravitationally bound together!
It is possible to use a solar sail to steer the Earth into the Sun. Just use it to tack against the direction in which the Earth is travelling, gradually slowing its orbital velocity and increasing the orbit's eccentricity, until the orbit passes within the Roche limit where the Earth is torn apart by tidal forces.
Billiards method. Clonk the Earth with something big and heavy, causing it to alter course. Let me emphasize the words "big", "heavy" and "clonk". Ceres, the solar system's largest asteroid, has less than 1/40,000th the mass of Earth; the Moon, a mere 1/80th. These objects are the heaviest you're likely to find - there are heavier moons and entire planets you could consider using, but to be honest from this point of view it looks more like using a succession of hundreds, thousands or tens of thousands of smaller asteroid impacts would be a better bet. Certainly, no single impact is going to do all the course changing you'll be wanting to pull off.
Note that the Earth does not and will not behave like a solid, rigid billiard ball under such huge impacts as these. For example, an object the size of Mars hit Earth once in the dim and distant past. Rather than simply bouncing off, the object destroyed much of both itself and Earth, causing a VAST spray of matter to be hurled off from the impact point; this matter coagulated into what is now the Moon. Basically, the point here is that modelling impacts like these is a tricky business. Do your numbers carefully.
Gravity assistance. This is a method originally proposed as a means of moving Earth to a higher orbit around the Sun in order to save it from the Sun's inevitable Red Giant expansion. It involves asteroids, like the above method, only instead of direct impacts, this time we just steer them past the Earth, allowing rock and planet to exchange a little momentum, with the result of an Earth moving on a slightly different track and an asteroid moving on a significantly different one. You could reuse the same asteroid again and again, looping it around a few gas giants and back to gain lots more kinetic energy from those gas giants in the same way that Earth just gained velocity from the rock. You could repeat this thousands of times over the course of millions of years. Better, you could use many, many asteroids one after the other in a steady stream, and cut down the total time significantly. You could of course use this method to steer the Earth in any direction you wanted, not just away from the Sun... heh heh heh...
Ways of moving the Sun
Since the Sun carries the vast majority of the entire mass of the solar system, any force which moves it is likely to drag all of the planets along with it. This can be employed to move the Sun and Earth in tandem to a place where the Earth can more easily be destroyed.
Moving the Sun is about 6 orders of magnitude more difficult than moving the Earth but the Sun is continuously emitting energy which can be productively harnessed for this purpose.
Shkadov thrusters et al.. Build an enormous lightweight "hat" for the Sun, which catches the Sun's rays. This structure would not be in orbit around the Sun, but static, remaining aloft using the radiation pressure of catching and reflecting solar energy. If balanced correctly, the "hat" neither falls into the Sun nor is blown away. With half of the Sun's radiation blocked/reflected in the opposite direction, the Sun now has a net thrust upwards (i.e. in the direction of the "hat"). This results in a minute theoretical acceleration but over millions of years the velocity would accumulate to something substantial.
Ways of not moving the Earth
"Give me a place to stand, and I will move the Earth", proclaimed Archimedes upon discovering the principles of leverage. Unfortunately, that place needs to be immovable, which is impossible. Also, what he didn't specify is that he also needs an equally immovable fulcrum - equally impossible - and a lever of stunning length which is unbreakable. Hah! Nice try, Archimedes.
Okay, folks, let's look at this. Suppose everybody on the planet weighed 100kg (which is an overestimate, 70kg is more like it, probably less). Suppose that there were ten billion people (another overestimate - there are about 6.4 billion at the time of writing). Suppose that they all jumped ten metres in the air (a huge overestimate, fifty centimetres is more likely and probably much less). Suppose they were all at the exact same point on Earth (which they won't be, thus mitigating the effects of the jump). And lastly, suppose that they all jumped at precisely the same instant, which of course they will not, seeing as the time difference between the fastest watch and the slowest will likely be over five minutes.
Altogether that's a mass of one billion tonnes of humanity jumping ten metres in the air.
The Earth has a mass of... let's be nice and round it WAY down to 1021 tonnes. That's a trillion times heavier than all of humanity. Which means the distance the Earth moves when everybody jumps will be one trillionth of the distance that all the people jumped: that is to say, 10-11 metres, or about half the radius of a hydrogen atom.
It gets better. Even assuming the Earth did move by some significant distance when everybody jumped, just think about it: it'd move right back again! You jump up, the Earth goes down: you fall down, the Earth comes up to meet you. Jumping up and down to try to move the Earth is like mounting a fan on a sailboat, pointing the fan at the sail, and expecting the boat to move forwards. It just doesn't work!
Written by Sam Hughes, contributions from Adrian Filipi, Colby Hayward, Nate Johnson, Andrew Kirkpatrick, Jim Klein, Jasper Spaans, Doug Sundseth and Simon Tatham.