What can scientists do to keep an asteroid from hitting the earth?

Yes, that is an option. Any force applied to an object several millions of miles from Earth will be magnified in its effect - a small deviation will result in a very large deterrence. Another option is sending a heavy satellite to orbit the asteroid; the gravitational pull from the object will cause the asteroid to deviate from its course.

There is a great show by Dr. Phil Plait on the Discovery Channel called Bad Universe. That was what he covered on his first episode. :)

There are many problems with "nuking" it or trying to blow it up. Mostly the debris left over, which would still hit the earth. Forget what you saw on Armageddon or Deep Impact.

The most reliable and technologically feasible approach is the Gravity Tug (basically park a mass near the asteroid, and keep moving it about).

A nearby explosion could deflect it too, but we would need to know what kind of body it is (rocky, metallic, conglomerate).

There is also the possibility of using a solar reflector to either vaporize parts of it creating some sort of thrust (not good for rotating bodies).

What scientists really need is more money to work on this. Sadly, the people who control the money are scientifically illiterate stooges bought by corporations... Sometimes I think we deserve to be wiped out like the dinosaurs...

The farther an asteroid is from earth, the easier it will be to deflect it with something like a nuclear missile. The problem with that, however, is not just getting the missile there, but being able to accurately plot the asteroid's exact course. Neither is possible at this time. Most likely, when we realize an asteroid is on a collision course with earth, it will be too late. I know that's pessimistic, but the odds of an asteroid collision are exceedingly rare.

It depends on how much warning we have. There are all kids of cool sci-fi sounding proposals.

Some of them include things like landing jet thrusters on the asteroid to steer it off course; but really were mostly at the mercy of chance and long time scales. We're not really prepared.

There is a table that charts the size of the asteroid with the necessary warning time, and there's a whole area of the chat that's red, in that zone are asteroid above a certain size for which the warning time was insufficient. In those cases. We're f-ed. Extinction event. Nuclear winter.

There are more people working in a small town McDonald's than there are employed to seek out and document asteroids coming near the earth.

No, they can't blow it up because that would actually make the situation MUCH MUCH worse. It is better for a single mass to hit the earth than to blow that large mass in to millions of tiny pieces that can spread across the globe and destroy large swaths of the planet.

One weird solution is to position an ion-drive spaceship nearby and use the gravitational force between them to tow the asteroid. Would take a while, I think, but no problems with trying to push something like a comet made of irregularly-shaped fragile ice.

Or position big solar mirrors to cause outgassing on the surface and generate a bit of thrust that way.

If detected early enough, giving it a "gentle nudge" would be the most effective. Blowing it up would generally compound the problem. Instead of one huge impact causing tons of localized damage and minor-above average global damage, detonating an asteroid would cause tens of thousands of smaller impacts, probably with a smaller death total, but defintely much more widespread devastation.

Pray. At present there is no system in place to even detect an asteroid let alone redirect or blow one up. I have heard a detection system is coming and NASA is considering how to deal with the issue.

Knock it off course with nukes is a practical proposition if they get enough warning.