Playing music by ear is very useful possability. Almost every composer, or musicains in ganres like Jazz can play songs by ear, because it's very helpful.
Did you found yourself in situation, when you are found some interesting song, or riff, maybe solo, but you cannot found tabs or note sheets? Well, i think it's not so usual problem, but anyway, it's a lot more interesting, to have some adventure, and trying to understand more, how music works, without help of anyone.
Honstly, when you make your first attempts, it's feels very difficult, because you never do something similar before. But, with time, it's become a lot easier, so in some situation, you can found right notes in seconds, or maybe minuites.
The only thing you need to know about music, it's to compare two notes. Like you hear A note, and then you play G note on piano or guitar, and you should say, is you play note that lower, equivalent, or higher.
Now, try to practice a little. If you are have two instruments - this is great, but if no, you are can download piano simulator for mobile phone. Like Tonality.
And, now take some random note on piano, and remeber this note. Then, take any note on guitar or other instrument. If you are confused, a little, you can repeat this note as much times, as you want. Give yourself a question, you play the same note, higher or lower? Then, depending on your answer, you can check note you choose, or move it higher, lower.
Then, repeat the same, but now play random note on guitar, and try to found this note on piano. This is very good exercise.
After you practice some time, you can start to learn songs by ear. Let's take something easy, like Enjoy The Silence by Depeche Mode. To hear guitar better, we are listen to isolated guitar track here. Download or slow down this track, and split it by notes. Now, let's try to found the first note. You can try to find it by yourself, but if it's feeling hard, it's note:
Now, write this note in notepad or somewhere, and forget about it, go to another, asking yourself the same question, and move by semitenes. Now, just play the first two bars.
Now, when you are play you first song, maybe not perfect, but you can come back to this song at any time. In this situation, we just move nearly our root note. So, i have a little list for you, to practice more. I hide the first note by spoiler, if you are hard to found it:
After you practice a little, you can start to learn songs with more difficult intervals, than Minor Second (semiton) and Major Second (tone). You are need to train you "Interval Ear". I think, the best way to demonstrate it - this video. It's really great example, that also help me.
All other is just practice, so i also have songs for interval training:
Traditional Songs is a great website with a lot of songs to play. You have audiofiles, so you can download them, and listen to, and then check notes with tabs and note sheets included.
Good Luck! :D