The lvextend command you executed would only extend the filesystem by 800MB (assuming default extent size), a rounding error when you're looking at TB filesystems. The -l flag means "extents" which, by default, are 4MB in size. If you wish to grow the filesystem by, say, 200GB, the command would be: lvextend -L +200G vg2/lv_backup. (Note the difference between -l and -L.)
The problem is that the initial filesystem was 1TB (almost) and the new one is 800GB, so the new logical volume should now be around 1.8TB. From here, you can see that both the drives are part of the same volume groups:
The initial filesystem was 1TB. The new disk is 800GB. You added that disk to the VG, then used 800MB of that disk to extend the LV, then grow the filesystem on the LV, leaving you with 1.0008TB in the LV. Then you grew the filesystem to fill the LV.
If you want to end up with a 1.8TB filesystem, do this:
lvextend -L +800G vg2/lv_backup
or:
lvextend -l 100%FREE vg2/lv_backup
Followed by the resize2fs command.
An AMI, as you note, is a machine image. It's a total snapshot of a system stored as an image that can be launched as an instance. We'll get back to AMIs in a second.
Lets look at EBS. Your other two items are sub-items of this. EBS is a virtual block device. You can think of it as a hard drive, although it's really a bunch of software magic to link into another kind of storage device but make it look like a hard drive to an instance.
EBS is just the name for the whole service. Inside of EBS you have what are called volumes. These are the "unit" amazon is selling you. You create a volume and they allocate you X number of gigabytes and you use it like a hard drive that you can plug into any of your running computers (instances). Volumes can either be created blank or from a snapshot copy of previous volume, which brings us to the next topic.
Snapshots are ... well ... snapshots of volumes: an exact capture of what a volume looked like at a particular moment in time, including all its data. You could have a volume, attach it to your instance, fill it up with stuff, then snapshot it, but keep using it. The volume contents would keep changing as you used it as a file system but the snapshot would be frozen in time. You could create a new volume using this snapshot as a base. The new volume would look exactly like your first disk did when you took the snapshot. You could start using the new volume in place of the old one to roll-back your data, or maybe attach the same data set to a second machine. You can keep taking snapshots of volumes at any point in time. It's like a freeze-frame instance backup that can then easy be made into a new live disk (volume) whenever you need it.
So volumes can be based on new blank space or on a snapshot. Got that? Volumes can be attached and detached from any instances, but only connected to one instance at a time, just like the physical disk that they are a virtual abstraction of.
Now back to AMIs. These are tricky because there are two types. One creates an ephemeral instances where the root files system looks like a drive to the computer but actually sits in memory somewhere and vaporizes the minute it stops being used. The other kind is called an EBS backed instance. This means that when your instances loads up, it loads its root file system onto a new EBS volume, basically layering the EC2 virtual machine technology on top of their EBS technology. A regular EBS volume is something that sits next to EC2 and can be attached, but an EBS backed instance also IS a volume itself.
A regular AMI is just a big chunk of data that gets loaded up as a machine. An EBS backed AMI will get loaded up onto an EBS volume, so you can shut it down and it will start back up from where you left off just like a real disk would.
Now put it all together. If an instance is EBS backed, you can also snapshot it. Basically this does exactly what a regular snapshot would ... a freeze frame of the root disk of your computer at a moment in time. In practice, it does two things different. One is it shuts down your instance so that you get a copy of the disk as it would look to an OFF computer, not an ON one. This makes it easier to boot up :) So when you snapshot an instance, it shuts it down, takes the disk picture, then starts up again. Secondly, it saves that images as an AMI instead of as a regular disk snapshot. Basically it's a bootable snapshot of a volume.
Best Answer
It sounds like you created a new device (probably called /dev/xvdc). You'd have to create a filesystem on that then mount it.
UPDATE: You need to understand the relationship between the block device, the partition and the filesystem. The block device is the physical layout of the disk. Partitions are a way to carve up disks into logically-separated chunks. Filesystems are where your files actually go, there's generally a one-to-one relationship between partitions and filesystems. You expanded the physical disk:
... But your xvda1 partition is still 10G:
If this wasn't AWS, you would just boot into a LiveCD and expand the partition. As it stands, there's probably no good way to expand that partition. Amazon does some magic with partitions and takes care of the tricky stuff when you create instances from your AMIs. You'll have to make an AMI, launch a new instance from that AMI, and specify the new size in the wizard.
Once the instance is running you'd expand the filesystem (again, not the same thing as the partition, which Amazon will extend for you magically).