For example, when I plug in an external hard drive disk enclosure via USB, sometimes the disk is automatically mounted in /media/.
For example, the disk might be mounted to
/media/13f35f59-f023-4d98-b06f-9dfaebefd6c1
or
/media/4668484A68483B47
What are those long IDs?
The longer number is called a Universally Unique Identifier (UUID). It is specified by the filesystem. There is a good Wikipedia article that gives more information on why and how it is used.
FAT filesystems are identified with a shorter ID (the "Volume serial number"). Wikipedia suggests it's actually a timestamp.
It was probably mounted by udisksd. Look in syslog. On my system, inserting a USB stick results in this:
Aug 6 15:17:08 ubuntu udisksd[2856]: Mounted /dev/sdb1 at /media/mp/A88B-3652
udisks tries to provide a unique name for each drive or partition. It searches through all the symlinks that udev creates under /dev/disk. For removable media, it looks at the symlinks in /dev/disk/by-uuid and /dev/disk/by-label. For nonremovable media, it looks in /dev/disk/by-id.
The entries in the by-uuid directory are often, but not always, UUIDs. In your example, the UUID 13f35f59-f023-4d98-b06f-9dfaebefd6c1 was likely generated by a command like mke2fs, which creates filesystems.
In the case of my USB stick, A88B-3652 is actually the volume serial number of its fat32 partition.
The entries in the by-label directory are Volume Labels on CD or DVD discs, for example "Ubuntu 14.04 LTS amd64", or filesystem labels created with the -L option to mke2fs or tune2fs.
The entries in the by-id directory are often of the form bustype-manufacturer_device_serialnumber, with a -partn appended for each partition.
