There's a lot of information available on learning by ear. Rather than try to duplicate it, I refer you to some useful sources of information.
One useful tool for learning tuns by ear is software that can slow the music down without changing the pitch. Obviously this makes a big difference in learning by ear. See below for a discussion of relevant software.
A lot of people have thought about how to learn by ear, and there's a lot on the internet to help you out. Rather than repeat all of it, let me provide you with some helpful links.
Here are a few sources that I've looked at and think should be useful. Many are oriented toward Irish music but should be helpful for other kinds of music too.
One really useful tool for learning by ear is software that slows down the music — but without changing the pitch. Things that are difficult to hear at full speed are often quite obvious at 75% speed. For some of the most difficult musical phrases in the tunes I transcribed for this web site i had to slow the tunes down all the way to 25%. In some cases I even had to set up a repeating loop with as little as one note to two or three seconds worth of music to figure out what was going on.
Of course when you're first starting you won't be looking for such detail; but still, slowing down the music really helps.
There is software available for Mac, iPhones, iPads, Windows, Android, Linux, and probably more. My preference is for an app that displays the music visually, usually in the form of a moving display of the waveform. This lets you use the mouse or your finger to scroll through the music; you can select a portion of the music visually, and with a little practice find phrases in the music. You can slow it down without changing pitch; but you can also change pitch to match your instrument. You can select an area on the display and set up a loop to be played repeatedly until you figure it out.
The best known of the choices is the Amazing Slow Downer. But it doesn't have the waveform display. Here are a few that stand out; there are many more that are very good.
Transcribe. This is on the Mac, Windows and Linux. It was the first I knew of to have the waveform display. It looks rather outdated, but it actually is very capable and many options are clearly visible on the main screen.
Capo. Available for Mac, iPhone and iPad. This is one of my two favorites. It slows down, changes pitch or even key, it can filter out varioius frequencies to bring out the fiddle or the piano, and much more.
Anytune. Available for Mac, iPhone, iPad. This is my other favorite and the one I use the most. It can do everything listed for Capo and lots more.
Once you've made some progress, be sure to find other musicians to play with. Go to a local jam session, or perhaps to a weekend or week-long event. If you live in the Northeast, Maine Fiddle Camp would be a great place to go.
Meanwhile, click on Next Page to begin Part II of this section, which looks as ways to combine learning by ear and learning from written music, and looks at the effects of your learning style on what you actually learn.
Learning Tunes (By Ear & Reading)