Ola Manuel,
Recording equipment depends on what your aim is. If you just want to record to verify, you will need little more than a recorder with good built-in microphone, and you will only record in mp3 compressed format.
A lot of people use remembird, with satisfying to good results:
http://www.remembird.com/
If you want to playback in order to get the bird closer (be warned, I am not a big supporter of this 'disturbance'), you will need an external microphone, such as Sennheisser 300 MKE:
http://www.amazon.com/Sennheiser-MKE-300-Microphone/dp/B00005UQIH
If you want high quality recordings, and are planning to study birdsounds and make sonograms of the recordings (this is quite necessary for some species to be accepted as rarities), I can advise you the (expensive) Sennheisser ME66 or ME67 (67 is a little longer):
http://www.amazon.com/Sennheiser-ME66-Short-shotgun-capsule/dp/B0003066X2
This microphone will only render the highest quality if combined with a good sound recorder, which can record in WAVE or PCM. But remember, those solid state recorders are expensive and recording in high quality requires a lot of memory cards. Another important aspect is the connection with the recorder. As you see in the pictures of Sennheisser MKE300, it will connect to a mini jack. The Sennheisser ME66 will connect with a XLR cable, which can give higher quality IF connected with a soundrecorder which also has that XLR port. If not, you end up with very high quality input at the microphone, which will be partly lost (and you will have more noise) at the connection with the sound recorder.
Don't connect simple voice recorders with your microphone. They have a low signal to noise ratio, and very low internal amplification of the signal that comes in.
I am perfectly happy with a ME66, XLR to Jack cable and a Tascam DR-1 solid state sound recorder. It proved to be sturdy and reliable, but has a rather slow starting up time:
http://solidstatesound.co.uk/tascam_dr-1.htm
One of the main advantages is that I can play sounds and scroll through them very easily with the big jockey wheel. The disadvantage is the jack input (instead of XLR), and the specific battery which can only be loaded at USB. AA batteries are more flexible. The one that is liked best at the moment is the Edirol R09HR:
http://solidstatesound.co.uk/edirolr-09HR.htm
You will find some more recorders and microphones through this website. Allmost all will fullfill your requirements I guess, but the very good ones don't come cheap.