Aquariums are difficult locations because of several issues.
The water surrounding your intended subject may be crystal clear to your eyes but there is still particulate matter floating in it. Also, water has a large light-bending effect and every inch you shoot through makes it more difficult for your camera to collect enough light.
When photographing fish on exhibit at an aquarium you are shooting through glass or plexiglass type material that is, at times, a full foot in thickness. While this is easy for your eye to see through, the light bending properties can make it difficult for your camera to focus properly. Also, flash reflects off of glass and leaves a nasty glare on images.
When you walk into an aquarium the displays all look incredibly bright to our eyes. That is because you are standing in a dimly lit room and your eyes have dilated. The exhibits themselves are not as bright as they appear to you. Also, your eyes are continuously recording light. Your camera only has a fraction of a second to collect enough light. This makes the lighting levels of the aquarium displays harder to photograph.
In order to take great shots at an aquarium you must learn to overcome the innate challenges of the location. To do this we must look at each challenge individually because each challenge affects the image to different degrees at different displays.
To overcome the challenge of water is to pay attention to timing and your subject's position. Plan to take photographs when the water is at it's clearest. Just before feeding time the fish are generally excited but the water is at its clearest and free of additional fish debris. During and after feeding the water is cloudy from food particles, extra air bubbles (created from frantic feeding), and fish waste. In addition to avoiding cloudy water, plan to photograph subjects relatively close to the glass. This will minimize the amount of light refraction from the water and particulate matter.
It takes a lot of strength to hold back thousands of gallons of water and large sea creatures rubbing against the glass. In spite of the exceptionally high quality of this material, it does still scatter light to some degree. Be prepared to switch to manual focus to make adjustments to your camera's autofocus if it has trouble locking on to your subject.
Your on-camera flash is not usually helpful with glass/plexiglass either. Because the flash fires directly ahead of the camera the reflection obscures the subject you wished to capture. To avoid this glare you need to increase the angle of the flash to the glass in relation to your camera. If your camera does not have an add-on flash option you can turn your camera (and yourself) at a 45 degree or better angle to the glass before taking an image. This can mean shooting upwards at a fish or shooting from the side. There can be some distortion of your subject with shooting this way but it should help avoid flash glare. If your camera accepts an add-on flash via hot shoe mount you can use a flash with a tilt head. By tilting the head of the flash at least 45 degrees you will move the position of the glare away from your lens.
Because exhibits often seem very bright to our dilated eyes, it can be confusing when your camera insists it does not have enough light. If you have a camera without shutter speed and aperture control you will need to rely on manipulation of your camera preset modes to make shooting adjustments.
If your camera has the ability to control aperture and shutter speed your task will be much easier. When taking photographs in an aquarium you should use a small F-Stop setting (large aperture) to allow more light into the lens. Shutter speed will then be set based on exposure values as determined by your light meter. Keep in mind that fish are almost always in constant motion and keeping your shutter speed about 1/60th of a second is a good idea. You will also need to set your film speed to a more sensitive number. ISO 800 is generally a minimum film speed for shooting without a flash while ISO 400 is probably ok with flash.
You can also make the most of the light available in a scene by photographing fish near the tank light source. Often this light source is at the top of the tank so you have to be careful not to photography only the underside of the fish when shooting upwards.
Also, when taking photographs in an aquarium, you must be mindful of proper photography etiquette. Most aquariums are very photography friendly (commercial photographers be sure to check for additional restrictions) but there are a few restrictions. Some marine species are very sensitive to light. Because of this, flash is not allowed at certain exhibits. Please take the time to look for "no flash" signs at an exhibit before starting your photography. Be respectful of other guests as well. Just because you REALLY want a great photo of a stingray does not give you the right to push anyone else out of the way or take the best viewing spot at an exhibit for an hour. If you really want a lot of camera freedom, consider an aquarium photo workshop offered by the aquarium. Some aquariums also limit the size/number of bags than can be brought into the facility and some ban tripods. Be sure to check ahead of time to find out whether or not you can bring your camera bag full of equipment with you. By following the rules of the aquarium you are visiting you will help to ensure that you (and other photographers) will continue to be allowed to photograph an amazing variety of fish without ever getting wet.