The error messages you got are because you ran the requested commands on partitions and volumes as well as disks. No problem. A disk is a physical thing; Linux calls them /dev/hda, /dev/hdb etc or /dev/sda, sdb etc. For convenience, disks can be divided into partitions, see http://en.wikipedia.org/wiki/Disk_partitioning
for more information. Linux calls the partitions on /dev/sda: /dev/sda1, /dev/sda2, etc.
A volume is what you create a file system in. The most simple volume is a partition; this is what you have in /dev/sda1 with your /boot file system.
More complexly, you can create a volume by configuring space from a pool of disk space taken from several partitions. This is what LVM does -- it creates logical volumes (LVs) from a pool of partitions assigned to it. In your case you have only one partition assigned to LVM.
The output from "parted /dev/sda print" shows that the /dev/sda disk is 8590MB and that it is all used, split into two partitions: /dev/sda1 and /dev/sda2. /dev/sda1 is your /boot file system (shown by your first post). /dev/sda2 is assigned to LVM.
LVM has been configured to create two volumes from the space assigned to it: LogVol00 and LogVol01. All the available space is assigned to these volumes; there is no spare.
Your first post shows LogVol00 is used for your / file system; it is gruesomely full at 99% (better to stay below 75% to avoid fragmentation and hence slow performance).
LogVol01 is used for swap; when your computer does not have enough memory to do its work, it temporarily frees some up by swapping the contents of memory out into the swap space on disk.
The only place you have to get space to give to your / file system without buying another hard disk is to take it from your swap space. You have 3.8 GB of swap space, probably way more than you need. Any idea why it is so big? There is no "right" amount of swap space; the "sweet" choice depends on many factors, including being desperately short of disk space!
Can you check how much swap space you actually use? Once you have decided how much swap space you need you could reduce LogVol01 and extend LogVol00 then resize the / file system.
None of that is going to be easy. Is LVM the right choice for you? It is powerful, flexible and permits snapshots which are great for backups but those benefits come at the cost of complexity, especially when trying to resize the / file system. You may be better off reinstalling without LVM.