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.
Unless you're using an EBS-backed EC2 instance the AMI image is extracted to a 10GB drive image that is recreated when the instance is started. I use instance-stored EC2 instances instead of EBS-backed ones for all my server instances. I then simply make the EBS volumes whatever side I need and mount them as secondary drives.
With the EBS volumes I found that using the xfs filesystem and simply using the entire EBS volume without any partitioning was the best course of action. To increase the volume when it was necessary I would perform a snapshot of the existing EBS volume and then create a new larger volume built from the snapshot. You then simply detach the existing EBS volume and attach the new one. Once the new volume is mounted it will show as the current volume size until you run the xfs expand utility command that has to be ran while the filesystem is live. Checking the capacity after that is done will show the new larger size.
Now if you're using an Ubuntu AMI you can install the ebsmount package and create a hidden directory on the EBS volume and configure the system to actually automount using udev when the EBS volume is attached to the EC2 instance.
Best Answer
If the data is fairly unchanging, put it in an EBS volume and make a snapshot of it. When you start each new node, have it create a new volume based on the snapshot and mount it. Making the snapshot is a fairly slow process, but creating volumes based on the snapshot is surprisingly quick!
If your data changes a bit, putting it into S3 is a simpler process to maintain and hundreds of nodes can pull the data at once without a noticeable degradation in speed (compared to just a single node pulling down data). Overall this will be slower than the EBS method above, but it'll be simpler to implement and maintain.