The subject of an image is incredibly important but it's easy in your concentration to overlook the background. While you're busy following a quick bird and snapping away in the hope that one shot will be "the one" it's hard sometimes to remember composition much less pay attention to what's going on behind your feathery model. However, the background will inevitably make you remember its existence after you take the picture as it makes or, more often than not, breaks the image.
GOOD VS. BAD
What makes a "bad" or "good" background though? A bad background is typically one that is busy and distracting whether through color, contrast, or sharpness. Whatever causes it to distract the viewer from the subject or anything unpicturesque would be seen as bad. This does include things like buildings, powerlines, and cars but also natural objects if they cause the background to be non-uniform
Good backgrounds come in three types: clean, detailed, and pretty. Clean is the common (though still difficult to archive) Christmas Card look: an extremely smooth, uniform background rendered so out-of-focus that it looks almost like a single color. Conversely, detailed shows much of the background and, by having the subject's environment in the frame gives a lot of context to the image. This one can be difficult to pull off and keep from becoming distracting but can look great if done properly. The happy medium (in my opinion) is what I call a pretty background. These show enough that it lets the viewer know what they are seeing but not so much it can in any way be distracting
TIPS & TECHNIQUE: HOW TO ACHIEVE A GREAT BACKGROUND
There are many parts to the recipe of a great background, some of them technical and gear related, others simply technique and field tricks
On the gear side of things, aperture is the name of the game. The wider the aperture (the smaller the number) the more the lens will be able to throw the backroad out of focus rendering it less distracting. This is actually one of the main draws to the more "exotic" telephoto primes. Though ridiculously expensive, they offer VERY wide apertures that aren't accessible with other lenses. Though lowering your aperture will make the biggest difference, there are several other technical things you can do. Sensor size makes a small difference, with bigger sensors rendering more blurry backgrounds (this is why small point-and-shoots struggle with this so much.) Focal length can also have an impact. The longer the lens the more blurry the background.
Gear is great but field knowledge is invaluable. There are several tips and tricks to get better backgrounds in the moment but the most important, and arguably the hardest, is just to pay attention. It's VERY easy to get distracted by other things and forget about what's behind the subject. Often all it takes to improve is just a step or two to the left of right, a slight change of angle, or a different focal length. Getting low also often helps, (in more ways than one.) Another good way is to utilize the distances from you to the subject and from the subject to the background. The shorter that fist distance and the longer the second the better off you'll be.
I hope this has helped you. The bottom line here is to pay attention to the background. As long as you actually think about it you've cleared the main hurdle. The rest is just practice.