There is a widely held belief that a full frame camera is "better" than a DX (also known as APS-C) one. To put it plainly, it isn't generally. So what are the differences? A full frame sensor is equivalent to a 35mm piece of film in size. It is also the same shape as 35mm, which is a little more rectangular than it is square. A DX sensor is a little bit smaller and a little bit more Square. Because it is smaller than a full frame sensor, it has a little bit less resolution because it is capturing less information. The larger the sensor, the more information it can gather, which generally means more resolution. From this explanation, the conclusion is simple - full frame is better.
But before you draw this conclusion, you have to ask yourself if you really need that much resolution? To answer the question properly, you have to consider what you will be doing with your images. If you only plan to view them on a monitor, even a gigantic one, you do not need a lot of resolution and therefore a full frame sensor is just not necessary. If you plan to print your images particularly large, then a full frame sensor may be helpful. If you plan to do ton of cropping, then having more to work with may be helpful. And lastly if you plan on doing a lot of post processing because you have tough lighting conditions or can't seem to get your exposures correct in camera then a full frame sensor may be helpful. By "large", "a lot" and "a ton" we mean LARGE and A LOT and a TON! Short of that a smaller sensor will be fine.
Also, before you take the plunge into full frame you must consider that not all lenses were formulated for the full frame sensor. Whereas you can mount a lens built for a full frame sensor on any DX camera, it will not work in reverse. So unless you already have full frame lenses, you are in for more equipment!
Lastly, a full frame camera will not make you a better photographer. No camera will. That's what we're here to help with!
Below is an example of the difference. The gray area is the DX sensor. As you can see the FX sensor is a bit wider. This is not a comparison of the actual size of the sensor but rather the difference in the aspect ratio. As you can see the DX image is a bit more square and the FX frame is more rectangular. Again this is putting any differences in quality.