fdisk -l shows the PC-style partitions on each disk.
cat /proc/partitions shows the available block devices.
sdb and so on are hard disks.
sda1 and such are partitions on these disks.
dm-0 and so on are LVM logical volumes.
pvdisplay reports the volumes that are LVM physical volumes.
df shows the mounted filesystems. If a filesystem is mounted from some place like
/dev/sda1, it's not on an LVM volume. If it's mounted from
/dev/mapper/group-logical, it's on an LVM volume.