Why can light pass through diamond but not through graphite despite of the fact that both are made of carbon? What makes a matter transparent? Is it dependent upon the crystalline structure?

For Diamond and Graphite the answer is simple: different electron mobility. Graphite is highly conductive and all conductors absorb and reflect photons of visible range since the fundamental properties of conductors: electron mobility. Since the electrons in conductors are, by definition, movable their excitation are not on an atomic level but on a lattice level following principles of collective excitations called "plasmon" (http://en.wikipedia.org/wiki/Surface_plasmon). (wikipedia)  So an electromagnetic wave (such as visible light) that hit an electrical conductor excite a surface plasmon that is an oscillation of charge, due to the displacement of the electrons of the conductors, corresponding to the oscillation of electrical field coming from the wave. Eventually the wave gets reflected and that is why most metallic conductors are mirrors. Diamonds have a different crystal structure, atoms are more far aparts, and thus electron mobility is reduced determining that diamonds are dielectric. The electromagnetic wave (visible light) that goes into the dielectric does not sparkle any collective response. This determine that only the photons with the exactly right energy (wavelenght, in other words color) to excite molecular or atomic single-electron excitations can be absorbed. The atomic spectrum of carbon is almost outside visible range (it has just a light band in the blue-violet region) and the molecular spectrum of diamond is far below in the infrared region. So all the visible light that goes into a diamond goes completely through it, except for few blue photons and the ones that steps inside the impurities giving the characteristic color reflections, (responsible for the http://en.wikipedia.org/wiki/Diamond_type, eventually proving that is a natural-borne one).

TL; DR     Yes, it is the crystal structure which leads to different C-C bonds, different allowed energy levels, and absorption of different wavelengths of light in the two structures. The other answers are good but this picture may make them clearer If you search Google Images for Energy Bands you will also images of how atomic energy levels overlap and broaden as atoms get closer to each other and form solids. Diamond is a wide band gap semiconductor, or insulator.  All the four carbon valence electrons are tightly held in bonds to the four near neighbors in the diamonds structure, as Eric Pepke describes.  The energy levels of these bonds make up the valence band in diamond.  All the available energy states in the valence band are filled by electrons.  The energy states in the conduction band are completely empty.  An electron (necessarily in the valence band) can only absorb light if it has enough energy to promote the electron to an available energy state - a state in the conduction band.  The diamond band gap is 5.5 eV.  That's a wavelength of 225 nm, deep in the ultraviolet.  The spectrum of visible light is about 390 nm (violet) to 700 nm (red).  So diamond appears clear and transparent in the visible.  (As Sambhav Karnawat said, diamonds might appear opaque if you 'saw' in the UV.) There is a picture here http://chemed.chem.wisc.edu/chempaths/GenChem-Textbook/Diamond-and-Graphite-927.html that shows the difference between diamond - where each carbon atom has 4 neighbors, each the same distance apart - and graphite - where each carbon atom has 3 near neighbors in a plane and a fourth neighbor in another plane.  It's the difference in bonding to the two types of neighbors that gives rise to overlapping valence and conduction bands, as in panel (c) of the figure, above.  These overlapping bands include empty states at essentially the same energy as the filled states.  Now, any wavelength of light (any energy photon,) even in the far IR, can be absorbed - and graphite appears opaque in the UV, visible, and IR.

Mostly what makes things transparent is that the electrons in the substance have energy levels such that the light doesn't have enough energy to knock the electron into a higher energy state. Diamonds do have a high refractive index, so they bend light a lot. This is why they are so pretty. You don't have to polish a lot of facets to get them to sparkle nicely. If you grind up a diamond, the dust stops a lot of the light or bounces it back for that reason. ETA: I'll answer this specifically for carbon, though the edited details seem to be going in a different direction. Diamond is a face-centered cubic crystal. It annoys me that they call it that, because it's just as good to say that it's tetrahedral, like stacking cannonballs, and I personally find that a lot easier to visualize. That's the tightest packing to tile 3-space, though it's surprisingly hard to prove (and also, it isn't the tightest sphere packing in a sphere, which is a big reason 3-D chemistry is so varied). So you can think of laying out balls in a flat grid as equilateral triangles. Then you pack another layer on top and below. Each sphere has 12 nearest neighbors, six on a plane, three above, and three below. Carbon has 6 atoms in the outer set of p orbitals. That works out nicely. There's exactly one electron between every pair of atoms, a place for every electron, and every electron in its place. That makes for very strong bonds, which is why diamond is so hard. Every imaginable place for an electron is filled, so in terms of quantum electrodynamics, it's very dense, which is why it has such a high refractive index. However, the electrons don't have a lot of other places to go. Trying to push one out gets in the way of the others, so it takes a lot of energy to move one. This is why it's transparent and is a good insulator. The electrons resist moving out of place, and neither visible light nor a battery has enough energy to do it. Graphite is basically slices of a diamond. They are just the planes, where every atom has 6 nearest neighbors. Since it's planes, the planes slide along each other, which is why it's slippery. If you mash it and get it hot enough, the planes stick together, making diamond. Almost enough, and it's a flawed diamond. A little bit, and it's like coal. The bonds in graphite are just the same as in diamond, but there are only 6 neighbors, so it only needs 3 electrons per atom to hold together. The other 3 electrons just sort of hang around and mill about. They are easy to push around. So graphite conducts, because a voltage can move them pretty easily. It's also opaque, because light can move them pretty easily, and the light is absorbed doing it. This is called a band structure, and it's responsible for how metals behave. Graphite is just a semi-metal, but it's the same basic idea. I've left a lot of quantum stuff out, but this is good enough for intuition. The problem with getting a general understanding from this one case is that carbon happens to be very nice and fits with geometry in an understandable way. Most atoms aren't like that. It is hellishly difficult to calculate energy levels. In fact, there is no analytic solution in the general case; the math is just too hard, so we either have to use huge numerical simulations or just be content to measure them and leave it at that.

the comparison should have been between black diamonds and white diamonds, However, without disputing the explanations above, there may be light in the form of single photons that manage to penetrate, but at such a level as to require extraordinarily sensitive instruments to detect them.

@Eric Has already answered the question. I would only like to add, crystal structure also plays a role. Eg. A diamond with an imperfect crystal Or inclusions in the crystal will not be as transparent. I would also like to add, transparency also depends on the visible spectrum of the observer. Eg. What maybe transparent to humans may not be transparent for a honey bee.

4 allotropes of the element carbon are known. One of these allotropes is graphite which is black in colour;  another allotrope is diamond, which is clear coloured. The other two allotropes C60  and graphene are lab created substances available in only very small quantities. I do not know their colour.  All 4 allotropes of carbon have different chemical structures, so will have different physical and ( sometimes different ) chemical properties. Colour is an example of a physicsl property. Graphite is the most common allotrope of carbon. It's also cheaper and easier to work with than diamond when using carbon as a laboratory chemical.  Industrial grade diamonds are used in industry as an abrasive material. Gem grade diamonds are sold in jewellery shops as a fashion accessory. Graphite has a wider range of uses.  Sulphur is another example of an element that exists in more than one form. ie it has more than one allotrope. It exists either as a yellow powder having chemical formula S8,  or a polymer of formula S6250. This is school chemistry ! What allotropes are, is covered in school chemistry texts books for 14 & 15 year olds. ( When I was at school at any rate ). Get hold of a chemistry text book for that age group. More advanced chemistry textbooks take it  for granted the reader knows what allotropes are.  Phosphorus is another example of an element than exists in more than one form. Eg white phosphorus, red phosphorus, & black phosphorus are all various forms of phosphorus, ie the element Phosphorus has more than one allotrope.  The element oxygen exists as O2, everyday oxygen,  and as O3 , ozone. ie oxygen has more than one allotrope. O being the generic symbol for oxygen; but oxygen does not ordinarily exist as O, but as O2 & as  O3. You cannot fill a gas cylinder with O. But you can fill one with O2  or O3. Likewise carbon does not ordinarily exist as C just by itself but as a sheet like structure of carbon atoms joined in a ring ( ie graphite ), as a tetrahedral lattice of carbon atoms ( ie diamond ), as a group of 60 carbon atoms joined one to another in the shape of a ball ( ie C60, mentioned above ), or as graphene; a newly discovered form of carbon by a group of researchers at Manchester University in the 2000's.  PS. Chemistry textbooks may express the reaction of sulphur with oxygen as  S + O2 = SO2 even though in reality sulphur exists as S8. They write the reaction, as above, rather than as S8 for the sake of simplicity.  Likewise the chemical reaction C + O2 = C02 as written in chemistry textbooks means: C ( graphite ) + O2 = CO2 or C ( diamond ) + O2 = CO2 ie. The authors just write C for the sake of simplicity, because the end product is the same, CO2; whatever  source of graphite or diamond, ie carbon, is  used for the chemical reaction. Writing down the chemical formula for graphite or diamond for what it really is : C1000000 or whatever it is, would make the formula look unnecessarily complicated. So chemistry authors simply write C; leaving it to the school teacher to explain what this form of short hand really means.  I've written oxygen as O2, oxygen's true formula, like in chemistry textbooks, rather than just O, because the chemical formula for the allotrope of oxygen we use in the lab and for breathing ie everyday oxygen, is simply O2. So for the oxygen bit of the above equation we write down that bit as it really is. But the carbon bit is too complicated to write down as it really is; so we just write down C as a shorthand. In the same way we write S as a shorthand for sulphur, because stating sulphur is in fact S8 isn't essential. The polymer form of sulphur S6250 mentioned above is not a stable allotrope of sulphur you can buy from chemical suppliers so there is no confusion writing down the formula  of sulphur as S rather than S8. ( S6250 is the molten form of sulphur you get as it comes out of a volcano during a volcanic eruption. S6250  then cools to form S8. S6250 can also be made in the lab simply by heating S8. It's a school chemistry experiment  ).

The nearest allotropic form of Diamond is Graphite, both of which are made of carbon chain molecule. But Diamond exhibit a different molecular struture than that of graphite. Graphite when subjected to high temperatures and pressures reaches at different stages out of which stage diamond stage is stable at normal conditions. Due to the allotrpic form transformation, they exhibit varying physical as well as chemical properties.