Doors – What kind of (1950s era) lock is this

This type of door is called a pocket door.

The first possibility is that the door has simply come off the track or the door has come unattached from the hanger that runs along the track. In that case you can figure out which side of the track it should be hung from and pick the door up and rehang it or attempt to fish the hanger out of the pocket and reattach it to the top of the door.

The more likely case, in my experience, is that the hardware is broken and will have to be replaced. Here is a good guide to replacing the hardware on a pocket door: http://www.ronhazelton.com/projects/how_to_repair_and_replace_a_pocket_door

If you look at the lower picture, you'll see that the missing tab served to hold the plug in its recessed (locked) position. When rotated 90 degrees it lined up with the cut-outs, letting the spring (lost, I presume?) push it out to the unlocked position.

It may also have helped retain the plug in the lock. More commonly, that's actually done by a reserved wafer, so the plug can be removed and rekeyed by cutting a modified key (or by picking). If you had a new key cut recently and the new key is a bit longer or otherwise not identical, you may have unintentionally engaged that function.

If you contact the window's manufacturer -- or the lock's manufacturer if it's an aftermarket addition -- they may be able to send you a new plug with instructions for installing it. If you give them the key number they can probably issue one that responds to the same key. If you can't find a brandname, a locksmith may be able to take measurements and track it down.

Note, however, that the lock cylinder is not required to operate the latch, only to make it tamper-resistant. So you could leave it as it is, or just shove the plug back into the lock and operate it as a manual bolt rather than one that automatically pops out when you turn the key. Not as secure as it was originally, but more so than leaving the cylinder out.