Huge topic, and I'll try to keep my answer as simple as possible. I don't do much in the way of coral photography but I do a lot of other fish photography and understand the basic alterations needed for corals. I'll start with general topics and then more practical thoughts.
The very first factor that needs to be looked at is the physical aquarium itself. When photographing fish and corals (unless you have a submersible underwater camera) you are having to photograph through several different material layers. The light has to bounce off of the subject (in this case the coral), travel through water, through glass, through air, through your lens, and finally hit your sensor. What is often overlooked is that for best results you need to shoot as perpendicular to the glass as possible. When you are shooting at an angle, the light is going to get distorted and your image is not going to come out sharply in focus. Our eyes and brain are great at compensating for such distortion, allowing us to instantly correct minor distortions, but the camera isn't. It's going to show exactly how the light hits the sensor, so we have to eliminate as much distortion as possible. This also means that for sharp pictures you must take your photos through a flat aquarium side. Bowfronts, Biocubes, and any other aquarium that has any curvature to the side you are looking through will be impossible to get clear, detailed images. Of course, you also have to make sure the glass is as clean as possible.
The biggest factor in any photography is light. The worst camera in the world can take a great photo under perfect lighting, and the best camera isn't going to be able to do anything if the lighting is bad enough. For fish, I strongly recommend flashes as you need the extra light to be able to freeze them and eliminate motion blur, but unfortunately the light that flashes produce isn't great for photographing corals. They're going to wash out virtually all of the color that we are looking for. We all know that blue lights make the color of the coral come out, so they have to be included in order to make the corals pop in the picture. Unfortunately, going with only blue lights is going to make the entire tank blue, which is less attractive, so you're probably going to want to have both the white and blue lights. The white light should make your rocks have a natural color while the blue/actinic lights will bring out the colors in the coral. This does bring us to white balance. We all know that there really isn't such a thing as "white" light. It all has some sort of tint, our brains just compensate to make white look white. Again, camera sensors don't automatically do this and record the light whatever color hits the sensor, whether it's a red tint, a blue tint, or any other color tint (though most light is either slightly red or slightly blue). White balancing is the process the processor in the camera (or the software in the computer) uses to do this same compensation. Most cameras have decent automatic white balance, but you can also play around with other presets to see what works best. If you want, you can manually set the white balance in camera in which case you will probably just be guessing and adjusting until you find what's right for you. Or, the best option, is to shoot in a raw format which only preserves the base data without this processing applied which allows you to manually adjust it in image processing software such as Photoshop or Lightroom. However, even with shooting in Raw, if the overall light color is inconsistent or completely wrong it's incredibly difficult to fix (and would require processing the image multiple times and compositing the result, so let's try to avoid that). For example, when shooting with only blue light, you are not going to be able to adjust the white balance to prevent the rocks from looking blue without also changing the color of the corals. Which is why we need to try to make the overall light as close to what we want to see in the final picture as possible. Hopefully I didn't ramble to much in that section!
In addition to light quality (color, etc), we also have to worry about light intensity. Camera sensors are much less sensitive than human eyes, so they require a lot more light in order to expose properly. This is why flashes sometimes hurt our eyes with their brightness yet expose the picture properly. The light in our tanks, comparatively speaking, is relatively dim (even the bright lights of a reef tank). There are three controls in camera to compensate for light intensity, and they each interact with the others and have noticeable affects on the final image (Google "exposure triangle" for many sites that will explain it in greater depth than I will here). The first is shutter speed, which is how long the sensor is exposed to the light. The slower the shutter speed, the longer the sensor is exposed, the more light hits the sensor, the brighter the picture (and vice versa with fast shutter speeds reducing the amount of light). The other affect on the final image is how much the subject is frozen. Slow shutter speeds open you up for motion blur while fast shutter speeds freeze the subject more. The second control is the aperture of the lens. This is how large the hole that lets light through the lens is. It's a little confusing to relate the numbers and what is happening at first, but it becomes easier to understand with practice. The aperture size will be numbered with such numbers as f2.8, f4.0, f5.6, f8, f11, f16, etc. The smaller the number, the larger the hole that lets the light through, and thus the brighter the picture. The larger the number, the smaller the hole, the darker the picture. The other affect on the final image is the depth of field, or how much of the picture is in focus. Larger apertures (lower f numbers) not only create a brighter picture but also result in a shallower depth of field. Smaller apertures (larger f numbers) are darker but will have more of the image in focus. For example, if you want just the face of a fish in focus while the tail is out of focus you should use a large aperture and if you want your entire tank in focus you should use a small one. (Distance also affects depth of field, but we'll mostly ignore that for now. Just know that the closer the subject is to the camera the shallower the depth of field while the farther from the camera the wider the depth of field.) The final control that affects the exposure is the ISO rating, which is essentially how sensitive the sensor is to light. The higher the ISO, the more sensitive the sensor will be, the brighter the picture. The tradeoff is that higher ISO's will introduce noise into the image, lowering the final quality. Cameras are much better now than they used to be in terms of high ISO image quality, but unless you have the newest, highest end cameras, you generally want to use the smallest ISO you can get away with.
In order to get a good exposure, we need to balance these three settings. What settings you choose will be based on the overall light level present, your equipment, and your personal choices. The automatic camera mode will set this for you if you want, but changing the settings can allow you to get the picture you want. It can be useful when just starting to look at what the camera sets and then adjust from there. For example, if you think the aperture is too large (because the depth of field is too small) then you can make the aperture smaller but then lower the shutter speed to compensate for less light coming through your lens. For example, I have a not particularly bright tank behind me. Automatic sets my camera to f2.8, 1/100 second shutter speed, ISO 800. The aperture is probably too large to get much in focus (particularly if shooting up close) so I would likely change it to f5.6, 1/30 second, ISO 800 which gives me a similar (very slightly darker, but it's ok) exposure. The one potential problem with this setting is that 1/30 second shutter speed leaves you extremely vulnerable to motion blur. Obviously if this is a hard coral that isn't going to move at all (like Acropora) it won't be a big deal, but any sort of coral with any motion will most likely blur. The best way to avoid this is to turn off all pumps in your tank for a few minutes before you start shooting (5-10 should be plenty). This will also allow suspended particles that may be floating around a chance to settle out of the water column. With the pumps off, coral motion will be very minimal and you should be able to get a shot with no blur. Just make sure to use a tripod because you likely won't be able to hold the camera still enough to avoid blurring due to shaking the camera.
The final piece of the puzzle that I will hit on is composition. Figure out what it is about your tank that you want to show off. It may be that you want a full tank shot, but most of the time closeups are going to be much more appealing. The more you try to fit into the picture, the more likely you are to have distracting elements that take your eye away from what you want to look at. If you have a specific coral you want to show off, try to zoom in to where that is the only coral in your shot. If there is going to be something in the background that you don't necessarily want to focus on (think stuff behind a gorgonian that you will be able to see through the branches), try using a larger aperture (lower f number) to keep only what you want in focus and blur what's behind it. Experiment with this and see how things change. For example, even keeping the same aperture setting, if you move your camera closer with a wider angle lens your depth of field will be shallower than if you back the camera up and zoom in to the same framing. A lot of the time, figuring out what exactly you want to show off can be the hardest part of getting a good picture.
So, to summarize and put all of this into practice, this is where I would start: Turn on all of your lights (both whites and actinics) so that the tank looks as close to the picture you want. Turn off your pumps and let things in your tank settle down a bit. Set up your camera on a tripod with the lens perfectly perpendicular to the flat front glass on the aquarium. Figure out how you want your exposure triangle to be set to get the picture you want. Take the picture. Experiment with your camera settings and composition until you get a picture that you like. That should get you through basic coral photographing 101. Of course, there are many more areas you could look at. Equipment can certainly affect your final results, but understanding and being able to control the basics will give you much better results than just buying more expensive gear. Once you get comfortable with the basics, there may be pictures you want to take that you need specialized equipment for. For example, some people want to maximize the fluorescence and cut back everything else and will use special lighting and filters to only allow the fluorescence to show. Also, virtually every picture you see from experienced photographers will have some sort of digital processing done to it. Adjusting color balance, lighting, contrast, etc. are important to getting the best possible final result.
