My suggestion (May 2016)
For the best capture record in the highest quality available, in below setup 96 Khz/24 bit: Røde VideoMic Me. It works with both iPhone & Android phones (probably other phones as well). It is directional so a lot of ambient noise is eliminated.
I have a Canon 7D Mark II which has audio with its HD Video. It can be improved with the gear suggestion below. Note however that the suggested Röde VideoMic Pro will not transform your sound recording in leaps and bounds, just cut out the ambient noise. You most likely need a parabolic reflector to capture high quality bird sounds.
The workflow cannot be easier. I mount the mic on my camera plug it in, boost the signal with the onboard switch, and then record the bird sounds as I film. If I only want the sound, then I will connect the Røde mic to the Olympus recorder and set it to 96 000hz/24 bit, or for the less picky people just record directly with the built in mics of the Olympus LS-14. NOTE! this is far from using a Telinga parabolic reflector, but much better than the built-in mic on the camera
Bird Sound Recordings
When you see the speaker icon on a 'Gallery' bird page it will indicate that a recording exist, and that by clicking the icon you will be able to listen to some typical sounds of the bird; e.g., song, or common calls.
If the page doesn't have the speaker icon then go and look for the recording at one of these great sites:
A beginners guide to field recording.
Review of different parabolic reflector mics
Visit Xeno-Canto and listen to some recordings and find out for yourself what sounds good to you. I liked recordings by Patrik Åberg and his Telinga/Olympus equipment choice is reflected in my suggestion (though he has an older Telinga mic).
Doug Von Gausig have shared his expertise of different sorts of sound equipment, and recording techniques on his site Naturesongs
Playback of bird sounds
Sibley Guides has a good article on the proper use of playback. In short avoid loud and prolonged playbacks. Avoid using it to lure very rare birds in breeding season where also other birders could potentially playback and thus increase the likelyhood of a spoiled breeding. The Proper Use of Playback in Birding