So, the other day I was dialing through 80 meters and heard a guy calling CQ, but he was pretty distorted.  After listening for a bit he said he was calling on AM, so I switched over to that mode and he came in crystal clear.  Tried to answer him but he couuldn't hear me.  My question is, how does one determine whether a contact is using AM instead of SSB?  And why would you use AM?  It seems to me that using AM would take up extra bandwidth and potentially interfere with more stations.

If your receiver is set to a USB mode and you encounter an AM signal you should be able to hear the (unsuppressed) carrier (if your receive frequency is even slightly off of the transmitter's exact frequency).

To tune it in properly, tune to the 'zero-beat' audio frequency.

These days, AM is generally used by 'audio enthusiasts' who are nostalgic about the sound of unprocessed or lightly-processed audio with a wider frequency range than is used for 'communications audio'. It will sound better to you on an SSB receiver (if you don't have an AM mode available) if you can open up the filter bandwidth to its widest option.

Am is really cool because of the fidelity of the audio. When the signals are booming its a lot of fun. I program am calling stations and listen for them a lot. A lot of old radios are AM only so its not used by a bunch of guys with modern equipment. I have worked a bunch of AM stations with my favorite being 10m band. Lots of west coast stations are easy contacts on AM when the band is open.

AM is popular among the boatanchor enthusiasts. There's some nice old sets out there and the sound can be almost broadcast quality. In fact, some of them run converted broadcast transmitters, naturally tuned down the the legal limit.

To properly receive it, put your receiver in AM mode and set your IF selectivity as wide as possible.

ETA: This is the inside of a Heathkit DX-100, one of the most popular AM transmitters of the '50s and '60s. At the back are the two final amplifier tubes and on the left are the two Modulator tubes and the associated tramsformers. The modulator had to be capable of putting out 1/2 the input power to the final RF stage. IOW, if the transmitter was capable of 250 Watts RF, the modulater had to be capable of delivering 125 Watts of audio.
There's a good reason they're called boatanchors.

If you have a panadapter (band scope), you can see it.  It will appear as a nice, wide, symmetrical signal on the scope (as opposed to a SSB signal).

It will stay there because of the constant presence of the carrier, whereas that SSB signal will hop up and down as the operator speaks.

Yeah, that makes a lot of sense.

1 QSO on AM, SC to AZ on my early random wire antenna.  Sound was amazing.  I remember the Op stating it was 'Impossible' for use to chat due to my station, yet there we were. #Magic of Ham Radio!!!

There is no advantage to using AM.  A transmitter that is rated for 100 watts will only be able to manage about 25 watts
in AM mode.  Further, most of the power is going into the carrier, not the voice modulation.  So AM, in the end, is only
about 10% efficient as compared to SSB.

Many of the guys using AM are on 160 meters running old AM broadcast band equipment.

Sometimes on 40 meters you hear signals you just can't clarify.  Often this is guys running old military equipment that
can only do Upper Side Band, whereas the convention on 40 meters is Lower Side Band.

Also, on 80 meters in the evenings you will sometimes here such signals, which are in Upper Side Band, and spoken
in coded Spanish.  Yep, you guessed it...

If you look at an AM signal on a scope, you will see a peak, which is the carrier, and symmetrical wiggly stuff on
each side.  Those are the sidebands, which are mirror images, and are the actual voice modulation.

If you filter out the carrier, throw away one sideband (pick one, there are only two) as you don't need both, they
are mirror images, you are left with a...  you guessed it, a single sideband.  It is all voice modulation.  No energy
is wasted on the carrier or another sideband which is not needed.  All the energy goes into the voice modulation.  
If you aren't speaking, there is no power output.

Once upon a time, AM was the only way to obtain audiophile quality audio, but only if you used "boat anchor" vintage gear, or built a so-called "Class E" radio. Such radios had very little in the way of bandwidth limiting features in the design. Using such gear results in spectral occupancy of 20 or even 40KHz. They are bandwidth hogs, and are the bane of the 80 and 160M bands when band conditions are crowded. By contrast, your standard issue Yaesu, Kenwood or Icom radio typically limits the transmit passband to a maximum of about 3KHz (which translates to 6KHz total bandwidth in AM because both sidebands are transmitted). Also, AM generally allows the low end of the transmit passband to extend down to DC, as opposed to around 100Hz on the average YaKenCom rig.

More recently it has been possible to obtain radios that can provide more than the "traditional" 100Hz to 2700Hz SSB transmit passband. Running SSB with larger than traditional passbands has become known as "extended SSB", or "ESSB". This is the modern and far more spectrally efficient way to obtain audiophile quality audio. A partial list of such radios is given here: http://www.nu9n.com/essb_ready_rigs.html. With the advent of this technology there really isn't any functional reason to run AM other than for nostalgia, or to flex your AM "muscles", since AM does put a strain on amplifiers and having the ability to run continuous legal limit AM marks the ownership of a powerful, robust transmitter.

One doesn't use AM for efficiency.