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.
The errors you got suggest that your file system is more recent than that supported by the kernel (most likely an ext4 on a kernel that only supports ext2/3). Some AMIs depend on a non-standard AKI (Amazon Kernel Image) to provide certain features.
In this case, the procedure you have followed seems correct, therefore, when combined with the error you received, there is a good chance that the AKIs do not match. Check the AKI used by the original instance, and compare it to that of the new instance. If they don't match, explicitly specify the AKI at launch time. (You can also manually create the AMI and in doing so, specify the AKI at that time so that it need to be specified at launch).
As an aside, a better AMI design will use PV-GRUB as a bootloader to load a kernel from the AMI itself (instead of requiring a specific external AKI), which simplifies the process of keeping the kernel up to date. The AWS documentation on user provided kernels is worth reading if you want to implement this in your AMI.
Best Answer
The snapshot process shouldn't be visible to the instance. The real problem would come when you try to restore from the "unclean" snapshot. If a process is writing to the volume (or rather, to an open file descriptor for a file on the volume), write requests in the instance's kernel cache may not be written to disk, or may only be partially written. These files would look corrupted on a volume made from the snapshot.
AWS recommends unmounting all volumes before taking a snapshot for this reason. http://docs.aws.amazon.com/AWSEC2/latest/UserGuide/ebs-creating-snapshot.html
(Obviously, you can't unmount / on a running instance. You could stop the instance to make the snapshot, but I would try to find another way to back that up. Presumably you only need mongodb configuration files on that disk, which are normally under /etc, /var, and /opt.)