Fileservers are very useful and often used to store people's home directories. Several computers can use the same fileserver to ensure that all of a user's files are available regardless of which computer they are logged into. If the fileserver let a user fill up all of the shared disk space it would prevent anyone from writing to the disk. This is generally regarded as A Bad Idea and so Linux allows a fileserver to only allow users to use a certain amount of disk space each. These are called quotas.
You can see what your quota is and how much you are using with the quota -v command.
pjh1003@tcmpc55:~> quota -v Filesystem blocks quota limit grace files quota limit grace /dev/sda8 2291368* 2097152 2306867 4days 4565 0 0
Disks mounted remotely use the `network filesystem', or NFS, and although useful using a disk like this is very slow. If you are running a program that writes a lot of output you should always try to write the output to a local disk, and then copy it to the NFS disk if needs be.