Audacity has a feature for doing exactly that.  There are tutorials around, like this one: http://www.podtopia.net/gettingstarted/removenoise.shtml

Well its all about the levels of the microphone of which you record into. Someone really needs to know that what goes in must come out and the whole process of levels in the microphone and the sound going in. I sometimes use , if I have some background noise, a dsp effect called an expander. What it does is it takes the lowest sound in the recording and lowers it further.

also, when the recording is done lets say you have low roaring winds or some rumbling sound.  You can also Eq the low ends of the exact frequency down a bit just so its not as noticeable if it doesn't distorts the high ends of the birds chirping. Me personally I wouldn't recommend using audacity;s noise removal. but if thats all you got, thats what you got.

Audacity's noise removal is somewhat crude yeah, but I've been using it with varying success. It's best for continous noise, like electronics or fan humming. I haven't got as much knowledge as Brandon, so if you can follow him, it'll probably be for the better. ;)

I mean that's why they say You are only as good as your weakest link. You can have a 1200 dollar microphone but if you record in a horribly acoustic isolated room then you will sound like a horribly acoustic isolated room. But for outsides like you said recording birds it will also have to do a lot of microphone used and the characteristics of it. If you want to record a specific bird you don't want to use a omnidirectional microphone and pick up everything around it. When you want to record  a specific sound your best bet would be too use a cardioid microphone. whether its dynamic or condenser depending on what frequency response you want. So it will only capture in higher decibels what you are pointing at while the background noise is lowered completely. Then you can use an expander in that situation to decrease it even further.

Brandon, just out of curiosity: for something highly directional and distant (like a bird), wouldn't a shotgun mic be best?  Or are they just too expensive/specialized to consider?

I have never wanted to actually record just one bird since birds always move around. but they are not expensive or specialized enough. They are made to record what you want. you can buy ones that are $100.

http://www.mediacollege.com/audio/microphones/directional-characteristic...

Here is a photo of microphone directions. Shotgun mics are usually spercardiods/hypercardiods if you look in the picture. They capture this pretty decent small area in front of the mic. but since birds are always outside, doing things, and amongst trees and all this stuff that makes noise. If you get lucky a shotgun mic/cardiod is all I can consider to pick up just one specific sound.

http://www.thenihility.com/random/birds.mp3

I uploaded a song of a bird that was actually recorded outside. its just so focused you cant really hear around it.

Have a look at parabolic microphones. There are examples of do it yourself umbrella/bowl parabolic microphones. Anyway, a good microphone and recorder is required.