Mac Minis initially were released with only 2 USB ports on the rear. These could only upgrade to Mac OS X 10.5.8.
If the Mac Mini has a CoreSolo, or CoreDuo processor, those could be upgraded to 10.6.8, provided they were upgraded to a minimum 1 GB of RAM first in their chip memory.
If your Mac Mini came with a Core2Duo, Core i3, i5, or i7 processor it can be upgraded beyond 10.7 as long as it has a minimum of 2 GB of RAM.
For more on the limitations of each operating system, see these tips:
https://discussions.apple.com/docs/DOC-2275 - for 10.5 to 10.5.8
https://discussions.apple.com/docs/DOC-2455 - for 10.6 to 10.6.8
https://discussions.apple.com/docs/DOC-2465 - for 10.7 and higher
Note, Mac Mini Servers may have diffierent limitations, and if you bought the server version, check in the Server forum for your Server version
to inquire about what upgrade you can perform.