Simply put, it's difficult to photograph small, moving fish in the typical low-light conditions of an aquarium regardless of what equipment you have. A lot of the time, patience may be more important than actual skill. This is especially since digital allows you instant review and you can reshoot if something doesn't look good to you.
Some of the photos I've seen at this forum were apparently done without a flash but I've personally not been successful with this. The low lighting in the aquarium usually necessitates a higher ISO setting which can make your photos grainy and unusable, especially with the usual compact cameras. With modern DSLRs this is less of an issue.
The key reason why I use a flash is because it helps to freeze action, otherwise a slow shutter speed with movement causes motion blur because the fish is always moving. Either that, or there will be camera shake because you are yourself shaking. Yes, you can put the camera on a tripod instead for stability, but then you are either confined to wait in the same spot for the fish of your choice to pass by or otherwise chase the fish with a tripod mounted camera, which defeats the purpose of a tripod.
Hence the use of flash. An external flash unit is better than a built-in one, and when it comes to external flashes, a flash on a bracket is better than one that is hotshoe mounted. To get a good quality DSLR (even an old one) plus a macro lens (allows for great results especially closeup, and particularly suited to photographing small objects) plus a bracket and an external flash will run you way over your budget.
For example, the Canon D60 is an excellent (albeit old) DSLR that performs excellent even at higher ISO settings, gives you the chance to use an external flash (perhaps a 2nd hand 420ex) plus a bracket...it is cheaper than getting even the newer Canon G series cameras, but the D60 alone will still go for US$200-300 on Ebay, depending on accessories and condition. And if you want a capable but reasonably priced macro lens, look into the Tamron 90mm f/2.8 macro, but once again, that can blow your budget even on its own.
So stick to a compact camera with zoom (and possibly a macro mode) for now. A lot of cameras on the market these days have such functions. Don't worry about an 'aquarium' setting. There's literally squillions of digital compact cameras out there so I can't recommend which one to buy. The next best thing is to look up the good photos here and get in touch with the photographers to see what gear they use.
Then, before you buy, compare all the options within your budget by reading reviews here:
I also wrote a tutorial here on shutter speed, aperture, ISO and composition in case you're interested:
And then hone your technique by experimenting and practicing so that you get better with composition, focus and exposure. I've only just started doing fish photography but have been doing general photography for little over 5 years now. Even so, fish photography is difficult, so don't get too discouraged or get hung up about the gear, because trust me, it's really not the whole story.
Once you get into photography though, beware. It's a dangerous and expensive hobby.