In order to cool off your ship, you have to somehow get rid of the excess heat. As the Three Laws of Thermodynamics go: you can't win, you can't break even, and you always lose. No matter what you do, you will generate heat that you must get rid of.
Radiators are pretty much the best method for doing this. A classic radiator is just a piece of metal exposed to the vacuum of space. This includes the hull of your ship. Heat will leave the radiator in the form of infrared radiation. However, this process is slow and requires you to have rather large pieces of metal jutting out from your ship. The benefit is that it is very simple and does not require you to use any other resources.
Another simple method of getting rid of your heat would be to concentrate the heat into some material with a high heat capacity, then eject it from your ship. This way would get rid of the heat more quickly, but the downside is that you would constantly be losing material. You'd have to replenish your heat sinks every so often.
As some of the other comments and answers have pointed out, you can also recycle your heat. Many spacecraft, like Voyager, Curiosity, and Cassini (RIP), used radioisotope thermoelectric generators (RTGs) for power. These converted the heat from the decay of radioactive Uranium into electricity. Some of the heat is also used to keep the spacecraft warm. Now, depending on the size of your ship, this may or may not be enough to keep it cool.
Your ship will radiate heat to space across its entire surface, regardless of whether or not you have radiators. Since surface area increases as size squared while volume increases as size cubed (commonly known as the cube-square law), the larger a ship gets, its surface area will increase at a lower rate. That means that if you have a small ship that only holds a few people, the heat loss from the body of the ship itself might be enough to prevent excess heat buildup. But for a large ship with lots of people and equipment, the heat will build up faster than it can be used to generate electricity and heat the ship. Even if you use some heat for electricity generation, that process is relatively inefficient, and you will always end up with more heat at the end than you had at the beginning.
So no matter what you do, for a large enough ship, you will have to get rid of heat. And there are only two ways to do that in space - by radiating it away or by ejecting hot materials.