Mac Minis from early 2009 on can run OS X 10.9 (Mavericks), the latest. It came with OS x 10.5 (Leopard) and the base model had just 1 GB of RAM. You need to upgrade to at least 2 GB of RAM (though at least 4 GB is better), and OS X 10.6.8 (Snow Leopard) before you can install Mavericks.
Etherenet; this is what mine has, if it means anything to you. With no telephone line to my apartment or wifi available, I use mobile broadband.
ethernet:
Type: Ethernet Controller
Bus: PCI
Vendor ID: 0x10de
Device ID: 0x0ab0
Subsystem Vendor ID: 0x10de
Subsystem ID: 0xcb79
Revision ID: 0x00b1
BSD name: en0
Kext name: nvenet.kext
Location: /System/Library/Extensions/IONetworkingFamily.kext/Contents/PlugIns/nvenet.kext
Version: 2.0.19
As others have said, the last Mini with a built in optical drive was the first generation of the present form, the 2010 model. It can run Mavericks.
From the Apple website:
Supported Models
iMac (Mid 2007 or newer)
MacBook (Late 2008 Aluminum, or Early 2009 or newer)
MacBook Pro (Mid/Late 2007 or newer)
Xserve (Early 2009)
MacBook Air (Late 2008 or newer)
Mac mini (Early 2009 or newer)
Mac Pro (Early 2008 or newer)
From 2011 on the Mini lost the optical drive, but you can connect one using a USB port. It makes sense; the reliability of optical drives is not up with other components. As an accessory, you can just unplug the drive and replace it if it fails, or transfer it to a new computer if it is still good.
Optical drive is becoming increasingly irrelevant, though it is still convenient, and could still be round for a while.
I do use mine to load (and save) any material that comes on CD (as a teacher, quite a bit), and still do watch DVDs occasionally. However, as a storage medium, CDs and DVDs do seem to be a bit wanting. They certainly don't have the multiple decades long life that was touted when they first came out, without even considering scripting and general wear and tear. With CD-R and DVD-R you are lucky to get 10 years…. 5 years is more like it.