What would be the effects if Jupiter was to blow up?
Note: Lumo's answer went up as I was writing this. His addresses what would happen if you turned Jupiter into a gigantic thermonuclear bomb. This is actually overkill. You only need to provide the gravitational binding energy of Jupiter (it's crazy to say "only" in this context, but nevertheless it is a six order of magnitude difference). This is the scenario that my answer addresses.
Blowing up the Earth is hard, and blowing up Jupiter would be roughly 10000 times as hard. If you don't get everything up to escape velocity eventually some or all of the planet will recoalesce. The explosion needs to provide at least the gravitational binding energy (simplifying that the density is constant throughout the planet):
$$ E = - \frac{3 G M^2}{5r}, $$
which for Jupiter is of the order of $5\times10^{26}$ tons of TNT.
Supposing you manage this the remains of Jupiter will expand in all directions in an incandescent plasma fireball. This wind will scour the surface the Jupiter's moons but otherwise spread rather harmlessly through the solar system. (I'm basing this on the fact that Jupiter's sphere of influence is much smaller than the typical distances between the planets, so the kinetic energy of the debris will be greatly reduced by the time it reaches any other solar system bodies. There is also the inverse square decrease in flux of course. Note: the other answers suggest that there would be enough radiation to heat the Earth appreciably. To really answer this properly you need some idea about how the explosion is going to work. I'm assuming a fairly "efficient" device which puts most of the energy into simply lifting the material out of Jupiter's gravity well.) The surviving moons of Jupiter would fly off into the solar system and orbit the sun in roughly the orbit of Jupiter.
None of the planets would be greatly affected in their orbit, however Jupiter has a long term affect on the orbits of planets and asteroids by exerting periodic gravitational "tugs." These interactions would stop, but it's hard to say what the long term impact of that would be without running simulations. But nothing is going to go flying off in a dramatic way - it would take a very long time for this effect to build up over many orbits.
The mass of Jupiter is about $10^{27}$ kg which, via $E=mc^2$, translates to $10^{44}$ joules. If one turned the planet into thermonuclear fuel in some way and detonated it immediately, about 1% or $10^{42}$ joules would be released. Because the diameter of Jupiter is about 130,000 km, the blast would last at least half a second or so. So we have $10^{42}$ joules per half a second. It's $2\times 10^{42}$ watts.
The Sun only releases $4\times 10^{26}$ watts of power, so the blast would be $2\times 10^{16}$ times stronger than the Sun. However, looking at the effects on the Earth, we must realize that Jupiter is about 5 times further from the Earth than the Sun, reducing the energy flux by a factor of $5^2=25$. So the half-second blast seems about $10^{15}$ times stronger than the sunshine. The equilibrium temperature is, because of the $\sigma T^4$ law, about $10^4$ times higher than that from the sunshine, about a million degrees.
The Sun warms the Earth by a degree in hours or so. A source that is $10^{15}$ times stronger obviously needs a tiny fraction of a second to reach thousands of degrees and evaporate the matter on the surface. So no doubt about it, the thermonuclear blast of Jupiter would burn and evaporate all nearby sides of all the planets – all of them are comparably far from the ground zero.
On the other hand, would the incoming energy be able to evaporate the whole Earth? We would be getting $10^{15}\times 342\times 4\pi \times 6,378,000^2\sim 2\times 10^{32}$ watts for half a second, about $10^{32}$ joules per the blast and per the surface of the Earth. The specific heats of materials are comparable to $1,000$ joules per Celsius degree and kilogram so we have $10^{29}$ kilogram-degrees to be heated. Divide it by the Earth mass below $10^{25}$ kg to see that you may still heat the material by tens of thousands of degrees by the incoming light. So I do think that this could evaporate the whole Earth but not the largest planets like Saturn.
Needless to say, the Sun itself would be pretty much untouched. Its surface already has 6,000 degrees or so. The strong radiation from Jupiter could bring it to a million of degrees, by the calculation above, but it's the same as the temperature of the interior layers. So the Sun would get destabilized a bit but it would quickly converge back to the Sun we know, I guess.
The calculations above are completely unrealistic because at most, one could think about turning Jupiter into a small star that would still burn very slowly and would be far weaker than the Sun.
Let's pluck several parameters from Wiki for blowing up the Jupiter. If we compare Earth with Jupiter in terms of mass and volume, it brings us a lot of wonderful numbers.
Volume: $1.4313×10^{15}\text{km}^3$ which can hold $1321$ Earths
Mass: $1.8986×10^{27} \text{kg}$ which is $318\ M_{Earth}$
It's indeed Sci-Fi. Because, simply using the mass-energy equivalence, we can see that it releases about $10^{45}J$, which is a lot of energy, indeed. First, the light along with the gamma rays from the nuclear explosion can easily cross 4 AU distance in about 36 mins to reach Earth. If we're planning to detonate the Jupiter's core, then we're gonna spread out its highly dense $H_2$ and $CH_3$ atmosphere, which can provide a medium for the shock-waves. Then, the highly radioactive radiations are gonna make their way to Earth within several weeks maybe. There will be a lot of consequences for this explosion like,
1) At the time light & gamma rays speed into Earth's atmosphere, we lose all our infrastructure on communications. Satellites in every orbit will be melted.
2) Next, we'll be watching jupiter on fire while we'd be slowly eroded (1 or 2 seconds, maybe) by the highly intense $\gamma$-rays, particles like neutrons and other EM radiation.
3) And, the most important thing. As I've said, the shockwaves would make their path through the spread-out atmosphere, taking the asteroid belt and some broken moons along with it.
Definitely, we won't be here to enjoy the fireworks ;-)