It's possible, as beeju has suggested, that you just need to manually mount the drive. However, you are correct--usually, Ubuntu will automatically mount a drive when it is attached or powered on.
There are two common situations where the problem you've described occurs:
Technically, drives are not mounted. Volumes (i.e., partitions) are mounted. If there is no partition on the drive, Ubuntu cannot mount it (neither automatically nor otherwise). If there is, but it's damaged or (this is much less likely) of a type Ubuntu doesn't recognize, it will also not be mounted.
Unetbootin requires that the partition to which you write the contents of an
.iso image be formatted FAT16 or FAT32. (And you should use FAT32 unless your image is tiny and you know you need to use FAT16.) Furthermore, it should be the first partition on the disk, and unless you really know what you're doing and know you need to do something differently, it should be the only partition on the disk and it should take up all the available space. Furthermore (though you don't usually have to do anything to make this the case), it should be a primary (not a logical) partition and the disk's partition table should be of type MSDOS.
If you have valuable data on your disk, then you shouldn't be writing an
.iso image to it anyway, because that would destroy the data. Otherwise, you can safely address either or both of the above problems by creating a new partition table on the disk, and then creating a single partition, taking up all the available space, of type FAT32.
One easy way to do this is with the GParted Partition Editor, which is provided by the
gparted package. You can install that package in the Software Center. Then run GParted. Select the drive in the drop-down menu at the upper-right corner of the GParted window. Make sure you've selected the correct drive--it would be bad to perform these data-destroying operations on the wrong disk!
Do Edit > Create Partition Table and accept the defaults. Then create a FAT32 partition, also with everything the default. This will make it a primary partition and as big as possible.
Then apply your changes by clicking the green check-mark. You can now mount the drive (or take it out and put it back in, and it should mount automatically). When GParted finishes, close out of it. Then you can write an .iso image to it with Unetbootin.
Alternatively, you might consider using a different method to write the
.iso image to the drive, which would let you skip over formatting and mounting the drive.
You can use Startup Disk Creator for this purpose, as described in the official installation instructions.
You can also use
dd from the command-line. See dd(1). If you have an
.iso image called
ubuntu-12.04-desktop-i386.iso and located in your Downloads folder, and if you would write its contents to
/dev/sdb1 in Unetbootin, then the corresponding
dd command is:
sudo dd if=ubuntu-12.04-desktop-i386.iso of=/dev/sdb
It's very important that you tell it to output to
/dev/sdb (or whatever drive you want it on), not
/dev/sdb1 (i.e., not the first partition on the drive).