Yes, provided there is an oxidizer .
A spaceship with liquid hydrogen and liquid oxygen will blow up quite well in the vacuum of space.
Chemical explosives will also explode in space since they function by breaking weakly bonded chemical components; no oxygen is necessary.
Nuclear explosions can of course occur in space, too.
However, in space, there is no atmosphere to transmit sound from the explosion to the observer, so it would appear to be silent.
The United States military in the 60's performed a series of nuke tests in outer space, and found out what EMP can do, when they wiped out Hawaii's electrical grid for a few hours.
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Explosions can happen, just not in the way that is depicted in the movies.
(this is how stars are created/destroyed, etc.)
Here is some nuclear testing in space.
For an example of an explosion happening.
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A ship which has lost structural integrety and is no longer able to contain the pressurized gas of its atmosphere will indeed explode.
The pressure of the escaping gasses would likely rip apart any structure that had been weakened.
In addition rapid expansion of gases from the heat generated by an explosion as well as the increased pressure from it would add to these forces.
For a small time the area around the ship would no longer be a vaccuum until the pressure from the atmosphere had been disipated.
During this time smoke and fire would be reasonably expected.
This probably goes on longer for the movie than it would in real life and more dramiticaly.
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The fire will most likely be extinguished in short order when exposed to the vacuum of space as the O2 will dissipate quickly.
A fire inside of a space ship will form a spherical shape until extinguished as shown at http://quest.nasa.gov/space/teachers/microgravity/MGprim1.html
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Yes, not all explosions require oxygen as a fuel.
Your standard flame requires fuel, space, and oxygen, but chemical fires can happen without the need for oxygen.
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If there is damage to their main power plant, it's possible for that system to explode.
Given that they're worked on by unsuited people, the power plants are surrounded by breathable atmosphere.
This could provide the fireballs we see, as the explosion consumes the ship's own life support.
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Flames are gases so hot that they glow and emit visible light.
Flames are heated by intense combustion - fire.
If the same gases are heated to the seme temperatures and pressures by other means,, they will glow the same.
Spaceships can explode from various causes in outer space.
Some types of explosions cold causes release of gases heated to the right temperatures to look like flames.
Some types of explosions could release lots of dust or soot, etc.
looking like clouds.
Some of the experts who answered before have said that e chemical explosions in a vacuum will not look like special effects explosions filmed in an atmosphere.
So fiery and smokey explosions in space are rather so-so in plausibility.
I personally prefer to see a blinding flash and then part or all of the spaceship vaporized and expanding from the site in a rapidly growing sphere which at first looks as bright as the sun but
as it expand and become thinner and thinner becomes less bright and begins to be transparent.
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Take for example the opening of  Revenge of the Sith . We see spaceships fighting with other spaceships using laser canons and missiles. Then when one spaceship is destroyed, we see fire and heavy black smokes coming from it.    Does that mean in space, fire can still burn, even though there is a vacuum?  If there would really be a space war, can spaceships really be destroyed and explode?
Can spaceships really explode in space?
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Spaceships can explode from various causes in outer space.  A spaceship with liquid hydrogen and liquid oxygen will blow up quite well in the vacuum of space. Chemical explosives will also explode in space since they function by breaking weakly bonded chemical components - no oxygen is necessary. Nuclear explosions can occur in space, too. Not all explosions require oxygen as a fuel. Some types of explosions cause release of gases heated to the right temperatures to look like flames, or release a lot of dust or soot, for example. It is possible for explosions to happen in space even in a vacuum and in the absence of oxygen: chemical and nuclear explosions for example. There may also be the appearance of what looks like flames and soot or dust. However, the explosion will not occur and burn like they do in the movies. Spaceships may also explode if they have lost their structural integrity or their main power plant is damaged.
Explosions can happen in space. These could be with oxygen, or chemical and nuclear explosion that do not require oxygen.  There may be some gases heated to the right temperature that look like flames and explosions tha release lots of dust, however, they would not happen as they do in the movies.  A spaceship that loses its structural integrity or that has a damaged main power plant may explode.