Welcome to AquariaCentral!
It may seem that way, but not everyone has a planted tank... I'm actually more incline to think that more people actually have tanks with no plants in them.
Plants can provide many benefits to a fish tank... as said by got_nailed. Generally, when comparing a healthy planted tank to a healthy non-planted tank... water in the planted tank is of better quality, because plants are in a nutshell, the ultimate filter. Conventional filters relay on nitrifying bacteria to convert ammonia to nitrite, then nitrite to nitrate. Whereas plants take both ammonia and nitrate and convert that to plant mass. Adding plants to a tank also increases the surface area of the tank, and thus increases the surface area for nitrifying bacteria to colonize as well. It is quite possible to run a heavily planted tank with nothing but a couple of powerheads for water movement (as long as the tank is stocked lightly and has fast growing plants).
If you decide to research further into keeping plants, you'll also notice that there are two kinds of plant tanks you can keep... the high tech tank, and the lower tech tank. High tech tanks have higher maintanence time, but also has rapid plant growth, and it is easier to acheive a "lush" aquascape in a high tech tank. Low tech tanks obviously have a lower maintanence time... but it plant growth is slower. Both have their pros and cons, it depends on what you want, and what kind of money you're willing to spend.
If you want the filtration benefits of plants, but you don't want to spend the time maintaing plants in a show tank, you can plumb your show tank via overflows into a sump. The sump can be planted with fast growing plants, and will give you the benefits of plant filtration.
By all means, you can keep a healthy tank with no plants. I keep a Malawi cichlid tank with no plants, and its inhabitants are thriving.
HTH
-Richer