Let me start with the ending:
Let me start with the ending because this is hard to discuss without being technical and you might get bored before I reach the point.
If you use a phone..
...to try to record a bird that's clearly audible to you but it's 1000ft away you will get home and it will sound like noisy garbage. No amount of cranking up the volume will fix it.
If you get a recorder built with the intention of capturing quiet sounds it's a whole different story and the recordings just sound amazing.
What I went with in the end is an Olypmus LS-7 from ebay for $80 plus I made a good fur covering (they call that a "dead cat") to eliminate wind noise. It works INCREDIBLY. If I really want to ID a bird I put on headphones with that thing in the field and it's like having sound binoculars. Plus it just sounds gorgeous. I did field recordings of just walking around old churches and crowds in Europe and it's epicly great.
If you just want to do that the LS-10,LS-11 etc are similar.
Now - Lets Look at How I Chose So You Can:
First let me mention some very useful statistics I found here all in one place. I'll refer to them later: https://www.avisoft.com/recordertests.htm
The gist of the problem with using a phone:
This approach tells you most of what you need to evaluate other choices:
First imagine the perfect recorder: if you were in a large room and there was something barely making noise at the other end you could hear it perfectly by just increasing the volume enough.
In reality that is not what you get because you run into two main problems.
1) Noise Floor: Recorders like your phone make noise on their own just recording silence. Let's call that the "noise floor". An analogy would be if you were whispering in a crowd - the crowd is noisy and your voice is below the noise floor. For the phone most bird sounds are below the noise floor - you hear the noise instead of the bird.
2) Dynamic range. A device and the file it is saved in has a limit to what range of volumes it can capture. If you were sitting next to a parrot (very very loud) and hearing a catbird 300ft away it's not easy to get both. What tends to happen is either the parrot makes the sound distort like a bad speaker or the catbird is just too quiet to get (or both). The wider the dynamic range the more difference between quiet and loud sounds you can get in a recording.
A phone has a low dynamic range and a high noise floor. It's not that the phone is cheap - it's really that it just doesn't need it - it's built to catch the human voice well. Very expensive microphones for music are similar - they expect to record relatively loud instruments.
If you were determined to find a way to use your phone you have a few options that are instructive for all recorder choices:
1) You can just plug in a better microphone. You can definitely get better but the best mic will still be disappointing because the electronics in the microphone port add noise - not just the microphone.
2) You can get a device that plugs into the phone that has it's own electronics - so you plug it into the port you charge the phone and send the data that way. That's a USB microphone. But basically what you've bought is a version of my Olympus that's limited to just connecting to your phone. And the point of the phone is generally that you always have it with you.... and you won't likely have the additional recorder. One reason you *would* want to go this route is to use software on the phone to manipulate sounds real time. I haven't tried it but you can plug the Olympus into phones to act as this so again it's the better choice most likely.
3) Amplify the sound going to the microphone - now you get into shotgun mics and parabolic mics.
A shotgun mic is a somewhat long tube that really tries to only get sound from the direction you are pointing it. The down sides are that it doesn't work fantastic based on samples I've heard, it's a huge thing to carry around, it's not likely to be cheap. A parabolic mic is very much like a satellite dish. It reflects all the sound coming from one direction into the center where you put a mic. My general impression is that shotgun mics help but they aren't amazing - plus you're carrying around a 2nd large thing - plus if you buy instead of build it will be expensive. I tried playing around with parabolic mics by just getting a golf umbrella as the dish and taping my phone in the right spot. It definitely did increase the volume of an aquarium across the house... but it was a heck of a lot to carry around, especially in a forest. And the Olympus just sounds better again.
What professionals use...
If we go beyond the realm of phones and a hand recorder you get into what professionals do - if you watch people doing work with film they will have a big mic on a poll, and it plus their headphones plug into a box hanging off their shoulder. That box is a high quality recorder probably costing thousands of dollars and the mics are probably very very good for the task at hand and also probably cost thousands of dollars.
Now if you look at that stats page again a lot of these devices are the expensive box I just mentioned, cheaper versions of it, and a lot are hand held recorders.
A useful example recorder
A very useful example is one that is popular right now with film and podcasting people: The Zoom H4 and the Zoom H2. They are a mix between a shoulder box and a handheld recorder. They have built in but you can plug in external ones. From stuff I've heard and opinions I've read the H2 does not have a low enough of a noise floor to work well for bids but the H4 is adequate. The H2 noise floor is -99dbu ein, the H4 is -107dbu ein, my Olympus is -122dbu ein. Those numbers are logarithmic meaning changing by a value of 1 is really a change in 10. So -99 to -107 to -122 are definitely big jumps you'd hear very clearly.
Another critical factor for field recording is wind noise.
If you are trying to pick up quiet sounds with a sensitive mic then almost any wind will sound like being hit with a jet of fire. Everybody has seen what they do on TV news, film, and TV sports for this. All these things are slowing down air around the mic. One option is foam around it. That helps but it has to be pretty still out to be enough. What just stops wind noise even in heavy wind is fur. They call that a "dead cat"... because it makes a mic look like someone stuck a dead cat on a pole. The smallest of these things is called a "baby ball gag", next is a "dead kitten", largest is a "dead wombat"- sound guys are odd. I wasn't impressed with the ones I saw on sale for mine so I tried buying fake fur at a fabric store. It worked better than foam but it wasn't great. So after reading how you need special fur for this with longer and shorter pieces I but a professional 'dead wombat' for pretty cheap and just cut the fabric out of that. It works perfect - the physics of it are impressive - to go from mild winds ruining a recording to high winds barely being noticeable. If you've ever worn a parka with fur around the face opening it's using the same principle - it stops fast wind at your face to keep it warm.
Sound file format
I'd call myself an audiophile but most audiophiles get nuts about tiny tiny differences in sound that if they can actually hear a difference in a blind test they are genetic anomalies. However when you're dealing with recording sounds you will amplify and dissect on the computer it's not how it sounds it's how much information you can pull out of nothing. Virtually everything you listen to is 16 bit (CDs, etc). If you have 24bit files you have more dynamic range to work with. Most recorders do 24bit - it is desirable.
A feature that's often missing in recorders but is pretty important. If you are out in the field you can record the entire thing for hours but most likely what happens is you'll hear a call, hit record... wait 2 minutes and hear nothing... then hear another call after you stopped recording. Pre-roll constantly records the sound put only keeps the last several seconds. So what that means is if you hear the call and hit record you'll get the beginning of the call. The Olypmus has 2 seconds and that is plenty. Might sound like a luxury but it's pretty life changing for recording birds.
If all you care about is recording for bird identification this isn't terribly important. When I recorded ambiet sounds in churches or if you record music this becomes important. My device makes great recordings and part of the reason is that it has a third mic that picks up very low sounds.
Functioning as a USB mic
Mine does this but I have yet to use it. It means you can hook it up to a laptop (or phone) and record live music for example. It is a great tool to have but I have yet to find a reason to try it for what I do.
Microphones are another topic
I could get into what I've read and experienced regarding microphones but if you're at that level you should be looking beyond my expertise.
Plans on a dead kitten
If anyone wants my plans for building something to hold the fake fur on an Olympus I might put an article up about that.
Part Two: Identification
I'm not so good at this. The birds I know I know. The rest - I can think of ways a database could be made to help you ID birds from recordings but I'm not seeing the resources out there. So I'll share what I can...
ID method one: Compare Your Guesses
The first basic way to ID from sound is to first get a vague idea of what you saw and use the call to distinguish. If you had a good recording of what looks like a wren in a photo you can easily go somewhere like allaboutbirds.com and listen.
ID Method Two: Memorize Calls - Verify Recording
The next is to memorize bird calls. If you could tell that sounds like either bird A or bird B - a recording is what you would want. I have bird call CDs and it just does not sink in for me. If I see a bird make the call in person - that seems to do it.
ID Method Three: Visual or Apps
A more advance way is to compare sonogram/spectrogram graphs. Here's an example from the wiki page:
The bottom part is what the waveform looks like in an audio editor. If you look at these for awhile you can spot what you hear by looking at a small enough group of them. If it's high note and three low - the graph will have a high line and three low. I've thought of writing software to somehow sort sonograms into things like how many notes, do any get low, etc. BUT - I couldn't find a good source of all the images and the number I'd have to get would be pretty large when you reallize many birds have more than one call. If you've ever used music identification software like sound hound, shazam, etc - that would work well to ID bird. There are a couple apps on phones that do it. The ones I've tried weren't so good but I haven't bought the ones that seem like they have better success.
ID Method Four: Ask Online
Apparently other people aren't great at this because if you go to for example reddit.com/r/whatstistbird with a photos they'll always have an answer unless it's a really terrible photo. With bird calls it's more like 50% chance.