Why is the universe so organized? If you think about the Big Bang and the flow of matter in all directions, you would think, how unorganized would this universe be? No matter how long it would take. The idea that matter or most of it will be organized in such a beautiful way is unthinkable. Check this for example.
The Moon orbits around our planet in a perfect fashion, not one but millions of system are like this. Our planet circles the Sun, again there are millions of examples like it. Our star circle the galaxy center in a predefined path. Again billions of stars do this! Now if you look at the matter that is organized (has found fixed orbit) vs that that is not, almost 90% of the matter is organized. They have fixed trajectories and have a stable path. In fact, if you want to look at matter that is not unorganized, you will find very little (we can think of our own solar system). Now there is not one galaxy but billions of them, all of them similar shape, like or spiral, which has stars. Outside this galaxy, there are few stars or maybe a supernova somewhere. 
Why and how is our universe so organized? 
Is the law incredibly simple, since the law has been repeated billions of times in every galaxy, or is it the work of some phenomenon of which the details are not yet known?
 A: Beauty is in the eye of the beholder. So is order.
What is a gigantic orderly mechanism for you, might be a horrible entropic mess for me. And we'd both be right.
A: A large part of the apparent organization of the Universe relates to the hierarchical and largely attractive nature of the four fundamental forces. The aptly-named Strong Force dominates other influences, so quarks are largely bound into baryons, which are neutral with respect to Color charge. The Strong Force also sees to it that baryons, when smashed together, tend to stick to each other in clusters, forming atomic nuclei.
The weak force mainly makes itself known with just occasional events like radioactivity. The electromagnetic force then attracts electrons to atomic nuclei, forming atoms. It also makes atoms tend to stick together, in molecules, and molecules stick together in crystals. Finally, gravity dominates once structures are electrically neutral, and is strictly attractive, except possibly on the cosmological scale, but cosmic expansion is not yet well understood anyway.
A: I would say that the universe appears organized due to three things:


*

*Newton's first law of motion

*Gravity

*Time


From your comments on other posts, you marvel at the straight lines and orbits, like fractals, that appear on all scales (moons orbiting planets orbiting stars orbiting galaxies orbiting clusters). This is due to the fact that all objects move in a straight line unless acted upon by an outside force. In astronomical scales that force is almost always gravity, and occasionally physical collision. The former force draws objects into either orbits or ejects them far from one another, and occasionally bodies collide to form a single, larger body that can now be even more gravitationally effective.
Of course, I am ignoring quantum, chemical, nuclear, and magnetic phenomenon here because I am trying to give a big picture. But if you try to imagine a disorganized universe, how would you describe it? Any description that I can come up with, considering the first two items that I listed above, will turn into a universe organized quite like our own given enough of the third.
A: To cite one the closest orbits to Earth, the moon is not in a fixed orbit. It is slowly spiraling out to space.
You might be interested in reading about the Lunar Laser Ranging Experiment
or the collision of the comet P/Shoemaker-Levy 9 with Jupiter. Even the Earth's rotation is not fixed. 
A: To some extent the universe exhibits something called self-organized criticality where a dynamic, non-linear system with many degrees of freedom (the gas after the Big Bang but before the emergence of structure) eventually forms a system with a notable degree of scale invariance (moons orbiting planets, planets orbiting stars, stars orbiting galactic centers, etc.)
I think another key aspect of this question is what do you mean by "organized"? According to some definitions the galaxy is hardly organized. You have stars here and there, random planets scattered about (both in and out of solar systems), you have random nebula and dwarf galaxies about. There are both great voids and weird wall structures. 
Another question is what sort of conceivable universe would be considered unorganized? Maybe stars frequently exploding, lots of galactic collisions all the time, etc.? Well, the reason we don't experience that is just due to the time scale. If we lived to be a billion years and our current perceived year seemed like a second, then that would be effectively what we would see. Everything would be pretty chaotic and fast changing. 
A: There are two elements to why the universe appears to be so orderly: the physical laws of that govern the universe are the same everywhere, and astronomical objects are very, very, very far from each other.
Consider two objects, one much larger than the other, and both very far from anything else. Because of gravity (which works the same everywhere), the smaller will either move in an ellipse around the larger, or fly by following a hyperbolic trajectory, and disappear into the distance. The exact starting position and velocity will affect only the particulars of the ellipse or hyperbola, but any set of starting conditions will result on one of those two.
Now consider the solar system. If you take something (much smaller than the sun) and stick it somewhere at random in the solar system and give it a random velocity, chances are very good that it will follow a hyperbolic trajectory or an elliptical orbit, because everything else is so widely spaced that it is unlikely to come close enough to anything else for it to matter much: the situation is very probably almost like the two-object scenario above, and the resultant path for the object is very probably almost what it would have been in that scenario.
Of course, the "very probably" and "almost" here are important. There are plenty of exceptions where objects pass near objects other than the sun, and you need to take the gravity of Jupiter and other planets into account to calculate paths to high precision. 
If you look at objects where no atmosphere is present to wear away the evidence (eg the moon, Mars), you see plenty of evidence that there have been plenty of collisions, and of course these collisions are still ongoing (eg Schumacher-Levy 9).
In systems that aren't so simple, such as star clusters or our galaxy, the situation is more complex. The main reason why galaxies and star clusters look so orderly is that the distances between the stars is so vast that, even when the stars are moving very fast, the change we see in their overall pattern over the course of a human lifetime is very slight. 
Even over longer timescales, though, the density is small enough that there are few collisions or even close interactions (see Binney & Tremaine's book, Galactic Dynamics, pp. 187-190). Instead, stars follow a potential due to the collective gravity of all the matter in the galaxy. In a roughly spherical system, this might result in roughly elliptical orbits, but might also result in rosette like (unclosed) trajectories (see Binney & Tremaine pp103-110). In a system were most interactions are gravitational, asymmetries in the distribution of distant stars are just as important as nearby stars.  (Although the gravity of a nearby mass drops as the distance squared, the amount of matter at a given distance increases as the distance squared.)
The specific orbits of stars within these potentials are not particularly ordered. You can see this if you compare the the smoothness in the distribution of young stars with those of old ones. Young stars tend to form in clumps ("star forming regions," places where there is a gas cloud under the right conditins for star formation, eg LH 95, IC 5146), so galaxies with lots of young stars tend to have visible structure (sometimes messy like I Zw 18 and NGC 4214, sometimes not, as in NGC 5248) the details of which depend on the dynamics of the gas in the galaxy. Over time, though, only the densest clumps remain together (because of their mutual gravity); otherwise, the variety of trajectories taken by different stars from the same star forming region will spread it out over time. Galaxies with mostly very old stars, like M 87, therefore, tend to be mostly very smooth, with a population of very dense clumps (globular clusters) which show up in our images as point sources because they are so far away. (Globular clusters in our own galaxy can be resolved, spectacularly, into individual stars; see, for example, M 13 and  M 3.)
Interestingly, the underlying randomness (disorder) in the trajectories of stars leads to interesting instances of apparent order. Just as the random motions of molecules in a gas allow us to use statistical laws to make precise descriptions of the behaviour of gas, the random trajectories of stars in globular clusters results in a surprising uniformity in their appearance. See this paper by Madsen. (Globular clusters are old enough and compact enough that interactions between individual stars can actually be important; see Binney & Tremaine p190).
On even larger scales, dramatic interactions between galaxies are quite common. NGC 3227 is a nice example; more can be seen here. In the case of smaller galaxies merging with the Milky Way, we can see the different trajectories of individual stars from the smaller galaxy spreading them out more smoothly over our galaxy. Several of these seem to be going on at once in the "Field of streams".
A: 
if you want to look at matter that is not unorganized, you will find very little

That's not true. Google for the words "messy room" and you'll get loads of images of unorganized matter. Even if you claim those messy rooms are the result of human, not natural processes, you can Google for the words "volcano eruption aftermath" and again you'll get loads of images of unorganized matter.
To make serious progress on answering this question, you need to define clearly what unorganized means. There is a physics concept that deals with this - it's called "entropy". However, entropy is not directly related to celestial mechanics. 
In a comment to one of the answers, you write:

Organized is any object that has trajectory and is not colliding.

Things definitely collide. The Moon likely formed from something colliding with the Earth many billions years ago. Meteors collide with the Earth every day (one of these killed the dinosaurs). And we're not unique, e.g. here's an asteroid colliding with another planet, two neutron stars colliding, and so on. Even galaxies collide (last I heard the odds are quite good that the Milky Way experienced such a collision in its youth). This neglects the microscopic collisions between atoms that e.g., give rise to atmospheric pressure, too.
If objects not colliding is your definition of "organized", then one can reasonably ask "why is the universe so disorganized". Because this completely opposite position is defensible, the conclusion is your question is too vague to be answered.
