I've been seeing these three formats on the internet. I remember a time when there was only mp3s. What makes flac and ape different from mp3s? Which format is best for music and why?

Flac - because of the better format. If you have good brand speakers you will hear the difference.

Both flac and ape are lossless compression algorithms, meaning no quality is lost on compression and decompression, unlike mp3 which is usually used as a lossy format. If your original audio is of high quality, flac and/or ape will preserve said quality. If your original audio is a phone recording of a concert, there isn't really that much of a quality to preserve.

Compare music encoded in both, if you can hear a difference, pick the one that sounds better. If you can't hear a difference, mp3 offers smaller file sizes. Good ears and good original audio is also needed. Flac encoding a shitty recording will still result in a shitty recording.

Flac and Ape are lossless audio format but not as popular as flac.
mp3 is lossy audio format having lower quality than other two.
If you keep increasing sound you will notice the difference else for lower sounds (i.e for mobile and ipods) you won't notice much difference.

Simply put, flac and ape are lossless formats. They preserve the original quality of the source audio from where they were ripped.
Mp3 is a coded / compressed format which results in loss of quality aka lossy compression. The benefit is an mp3 file can be multiples times smaller than a flac one. It's easy to fill up a hard-disk with only a couple dozen flac albums.
The truth is you probably won't be able to tell the difference between 320kbps mp3 and flac without headphones that cost as much as your computer so save the disk space and go with mp3 unless it's an album you really treasure.

Most people will probably not notice much difference between a good high quality MP3 and FLAC (or any other lossless format). But it's still preferable IMO to have FLAC because you can always convert it to MP3 yourself if you need a smaller filesize (say for your phone for example) whilst converting from MP3 to FLAC doesn't help because the information has already been lost.

But monkey can only use the Ape format :P

Monkeys and apes are not the same.

But Monkey’s Audio codec calls its files .ape, that will not help with the confusion…
Interesting read by the way, thanks for sharing it!

Except for the smell.

Your lucky the librarian is not here... (Discworld)

The only reason I haven't switched to FLAC or similar for everything is due to my personal set-up and limitations.

And I agree with anyone that would also raise the issue of how the files are mastered; loudness war "brick-walled" audio and such. Poorly mastered audio won't sound great regardless of the format you put it in.

I had the same thought. "Just don't say the M-word, sir."

Even with that, it's nice to be able to create a lower quality version of the tracks for the phone without having to deal with the additional degradation that happens when you convert from lossy to lossier.

Usually you won't notice any difference at all, when the MP3 file has been encoded with a modern encoder and a sufficient bitrate (> 128 kbit/s often is enough) and your ears are healthy, no matter how good your speakers or headphones are. There have been plenty of ABX tests showing exactly that, so there is not much point in arguing.
People with damaged ears are sometimes capable of distinguishing MP3 from uncompressed audio, because the acoustic model used for mp3 encoding assumes that the listener's ears have certain properties, which damaged ears sometimes do not have.

FLAC is still useful for archiving purposes. If you want to convert audio files, you usually do not want your source to be compressed in a lossy way.