So, how'd I do it? Well, I set my camera to a fairly large aperture of f/8.0 and I got as close to the fence as possible, while focusing on the bird's eyes. By doing this, the fence gets thrown out of focus, allowing the bird to become the subject. (I believe there was a railing between myself and the fence so I couldn't totally eliminate the fence in this photo, but you can see how it is greatly improved.)
What if you have a point and shoot camera and cannot manually change the settings? You will want to choose the auto setting that will cause the camera to change them for you.
Experiment with both the Portrait and the Close-up modes and see which one renders a better result for you. It may differ depending on your camera type and model. If your point and shoot has AV mode on it, try that, too. This mode allows you to set the aperture and the camera will set the appropriate shutter speed for you. Experiment with different apertures from f/8.0 and larger and see what happens.
The type of fencing will also affect the results. The eagle below was in more of a small mesh type of enclosure and it has almost been eliminated altogether, using this technique.
This technique is not just for bird photos, either. Try it with sports photos, too! Get out of the bleechers and down close to the fence if you can, after all, it's about your family member, right? Actually, I may have been in the bleechers when I took this photo of my niece, but again I used a large aperture and focused on my subject.
My physical distance from the fence is again why it's showing up as much as it is, but the shallow depth of field still allowed the fence to be out of focus enough to save the shot. This again illustrates why you want to get as close to the fence as possible. You still won't get a tack-sharp image, but you've at least minimized the obnoxious fencing and have a memory worth preserving.(f/5.0)
Practicing is a large part of becoming a better photographer, no matter what type of camera you use, (it's also the fun part!) so get out there and practice, practice, practice!