There are many movies with the Earth being saved by sending a bomb out blow up the asteroid. What realistically could we do to that degree today? Do we have interplanetary missiles? if so how much of an asteroid can we move?
There's no simple answer here.
In the simple case (no other encounters before impact) you need to generate no more (and maybe a lot less depending on where it's going to hit) than a bit over 6500 kilometers worth of deflection. Note the absence of time units.
Deflection, however will normally be measured in meters per second. Note that this does have time units.
Lets apply some numbers to get a better picture: You send out your bomb, it does a standoff detonation that imparts 1 meter per second to the offending rock. (Note that you do not want to blow it up in most cases!! Unless you can blow it to bits small enough that they won't make it through the atmosphere you actually make it worse.)
What does this do? If the rock was more than 6,500,000 seconds from impact that 1 m/s adds up to enough to cause a miss. That's 75 days.
Now, lets look at rock that is 10x as big. Lets send out the same missile. We only get .1 m/s deflection. Success means we need to intercept at least 750 days (just over 2 years) out.
Lets scale our rock up by 10x again. Now the same missile generates only .01 m/s and has to intercept almost 21 years before impact.
See how it's not a simple answer to how much asteroid can we move?
We have no such missiles sitting around, if a dangerous rock were detected we would have to build it, although not totally from scratch as we have already built something pretty similar--the Deep Impact mission. Replace the impactor with a nuke and include a rangefinder to set it off at the right point.
Also, you need to ensure you don't blow up the target. Since you can't be sure of what you're shooting at you want to make sure that the velocity imparted by the missile is well below the escape velocity of the rock so that if you break it it will put itself back together. (If you blow off small bits at high speed they'll miss anyway and don't matter.) This might necessitate multiple shots even if one could carry enough boom.
Since we have never shot at a rock with a bomb we don't truly know what will happen. NASA figures 10% of the energy of the bomb can be imparted on the target (this is basically an Orion drive.)
Note that this has one huge advantage over any other currently considered approach in that it's much easier to make a good guidance system than to match velocities with something going fast the wrong way. Lets look at the worst case--a comet going backwards. That could require 60 km/sec just to match and that's more than NASA could do. An Orion drive, however, cares not a bit that the warhead is closing on the pusher plate at 60 km/sec so long as your detonation timer is accurate to the sub millisecond range.