it checks and, if necessary, fixes the filesystem.
more or less, what windows does if you do not shut it down correctly.

fsck means "file system check" and vfat is for the dos file system (to make it short; there are others too, eg for ext3).
-a
means don't ask and fix automatically.
one needs to do that both as root and on the actual partition (/dev/mmcblk0p1).
normally, the partition has to be unmounted and fsck would complain if that's not the case ...

try to type in an x-term (does nothing to your system, just prints info)

mount

it will list the mountet partitions with their names, mount points and file systems, for instance:

/dev/mmcblk0p2 on /home type ext3 (...)

means
partition /dev/mmcblk0p2 is mounted to directory /home and has the fs type ext3, (...) is a list of parameters, like read/write and stuff.