Caveat - only having a desktop system not using LVM, I can't give you specific details here, but hopefully this will get you started.
Your system is set up with logical volumes
, which is a layer on top of the old style of disk partitioning. The advantage of logical volumes is that volumes can be extended (even across multiple physical devices) and moved in ways that are not possible with hardwired partitions.
The output of df is confusing because you cannot see the connections between the logical volumes and the underlying partitions. Have a look at commands like 'lvdisplay -m' and the other lvm tools.
Looking at the output of fdisk, the whole disk has already been partitioned and formatted, but not all of the physical space has been allocated to filesystems. I don't know how to do this for LVM systems, though you will find documentation here
On a server it makes sense to keep separate logical volumes for various purposes (whereas on a desktop the tendency is to preallocate all the spare space to /home). And because you can add logical extents to a logical volume without disruption, there is no need to allocate everything at the beginning.