In Synology terminology a disk group is simply a RAID array that can then be split it up into several volumes.
So if you have 4 disks, you can create one single disk group (RAID array) and create two volumes from it. Or you could create 2 disk groups, and create one volume on each.
The difference / advantages being that the single array will have more space, while the separate arrays will have better redundancy.
There is a nice write-up on the synology forums about this
SHR is a separate concept, and is based on the Linux Raid Management system.
It's a design that allows for the usage of otherwise unused space on larger disks added to an array limited by a smaller disk.
Full disclosure, I use synology at home but have never been involved with our NAS at work