You need to use the mount command. The following is an example for a device with at fat file system and would need to be run as root (or with sudo).
mount -t vfat /dev/sdb1 /mnt
you can use the dmesg command to when you connect the device to find out what the device file is (in this case /dev/sdb1).
Copying the files can then be done with the standard commands: eg "cp"
cp ~/Desktop/somefile.foo /mnt/
Once you have finished you should unmount the device:
Please read the mount man page, and/or google for tutorials for more information.