Chassis: SC933 (pretty sure this is it)
I almost purchased one of these for my first server and find them interesting. You don't see them very often, perhaps because they are EOL, but I find them interesting.
The 933 should be readily identifiable by certain "unique" aesthetic characteristics: (a) 15 vertically oriented 3.5" drives, (b) 3U chassis, (c) 3 PSU slots. I say unique as usually you will see up to 16 3.5" drives in a horizontal orientation (4 columns x 4 rows) and 2 PSU slots.
Here is the Supermicro link to your chassis:
3U Chassis | Chassis | Products - Super Micro Computer, Inc.
The below image is an example of a 933 and actually comes from this site, with a write up here:
https://www.servethehome.com/supermicro-sc933t-r760b-3u-15x-35-sassata-storage-chassis-review/
To your question, you can surely put a more modern board in there up to 12" x 13" E-ATX in size. As to what precisely, additional feedback is probably needed in regards to your use case, inclusive of what you hope to achieve with your upgrade.
As a preliminary thought, I might offer that if you have been managing with an X7 motherboard, a jump to an X10 and more expensive DDR4 RAM is probably not needed in your case. Moving to an X9 and less expensive DDR3 RAM would likely be a more optimal route to go.
Supermicro's naming convention was at first daunting to me as well; however, this resource explains it
Motherboards (Intel UP) | Product Naming Conventions | Super Micro Computer, Inc. and this matrix can be used to search
Motherboard Matrix | Support - Super Micro Computer, Inc.
