Your Apple "Monitor" [sic] is an iMac, and a full fledged computer like the Mac Mini is, except much newer. Now thankfully it does have Thunderbolt 2, which means you might be able to use Target Display Mode. But you need another thunderbolt capable Mac Mini for this. The way you can tell the difference that this is an iMac, is that it has an ethernet port. No Apple monitor had built-in ethernet.
Unfortunately the Mac Mini you have is also too old to be used as a boot system for the iMac, and the iMac is too new to be used as a boot system for the Mac Mini in Target Disk Mode having one Mac treat the other as an external hard disk. But it can be used for data transfer over Firewire to Thunderbolt adapters and the proper thunderbolt 2 cable. The Mac Mini will have to boot with the T key, and link to the iMac directly via those adapters once it is booted. It will be helpful to have a DVI capable monitor to see that the Mac Mini has successfully booted into Target Disk Mode before hooking the cables to the iMac.
Your best bet is getting a DVI capable monitor and USB keyboard and mouse to boot the Mac Mini, if its hard disk is still healthy, or you have a backup on a Firewire Hard Disk of its content, and networking the two machines via ethernet to transfer the data.
That Mac Mini is too old to boot via USB any hard drive, or any version of Mac OS X newer than 10.5.8. That iMac is probably capable of Mac OS 10.8 up to 10.15.