I need to run a dual 27" 2560x1440 mac-mini. I already have a thunderbolt display and I tried connecting a Dell Ultrasharp with 2560x1440 to the HDMI port of the mac mini - that doesn't work. There is a spare Thunderbolt port on the back of the Apple display, so do I use that and connect to the Dell? If so, what port on the Dell do I use to get the full resolution - HDMI, DVI, Ethernet?
HDMI is out because the Mac Mini's HDMI port maxes out at 1920x1200. You're going to need to grab an adapter to go from Mini-Displayport to either dual-link DVI or Displayport, which are both supported by the Dell.
Now, the Thunderbolt display can't directly output Mini-Displayport for boring technical reasons (see question 20). There must be a non-display Thunderbolt device between the display and the adapter I mentioned above. The cheapest example I've found is the Seagate GoFlex Desk Thunderbolt Adapter (STAE122), which has two Thunderbolt ports for daisychaining and retails for about $180.
The full chain:
Mac Mini -> Thunderbolt Display -> STAE122 -> MiniDP to DP or DL-DVI adapter -> Dell monitor