Every Mac with a Mini DisplayPort allows you to connect an external display or projector using an adapter. You can use an external display as your main workspace or to extend your desktop, or you can work in mirrored mode with a projector so you can view what your audience sees.
Use the Mini DisplayPort to DVI Adapter to connect an advanced digital monitor, such as the 20- or 23-inch Apple Cinema Display, that includes a DVI connector. Review the documentation or check with the manufacturer of your monitor to make sure you're choosing the right adapter.
Compatible with Mac computers with Mini DisplayPort and Thunderbolt ports
Thunderbolt is, depending on your Mac, either one or two 10 Gbps bi-directional streams in the case
Thunderbolt is, depending on your Mac, either one or two 10 Gbps bi-directional streams in the case of Thunderbolt 1. Thunderbolt 2 ups that to 20 Gbps. The MacBook Air is too thin for a full-sized controller, so it gets a single bi-directional stream, other Macs have dual bi-directional streams.
The Thunderbolt port is compatible with MiniDisplayPort displays. Otherwise Thunderbolt can provide audio, video, data & power to external devices like Apple's Thunderbolt display or various brands of hubs.
What I'm trying to say is that Thunderbolt is not a one-way connection but the visually-similar Mini DisplayPort is output-only.
I once connected a Thunderbolt hub to a MacBook Air & was able to connect four external HDs, one via Thunderbolt, one via Firewire 800, one via USB 3 & another via USB 2. This left a USB 3 port unused, so I connected a DVI display via a Display-Link USB adaptor. Everything worked fine, indicating two-way compatibility with HDs & one way with a display.
I suspect you have a HDMI --> Mini DisplayPort cable, rather than HDMI --> Thunderbolt.
Assuming you are using a recent Mac Mini with a HDMI port and a Thunderbolt port: Connect to one mon
Assuming you are using a recent Mac Mini with a HDMI port and a Thunderbolt port: Connect to one monitor using a standard HDMI cable. Connect to the second monitor using a MiniDisplay/Thunderbolt to HDMI adapter with an HDMI cable. Hope this helps!