External (Thunderbolt and USB 3) and Internal (PCIe) Capture Cards
Blackmagic Design offer a range of capture cards that connect to a computer via Thunderbolt or USB 3 or can be installed inside a streaming computer that supports PCIe.
On the lowest end of the price scale, their Ultrastudio Mini Recorder supports both HDMI and SDI inputs and is relatively inexpensive. It's a piece of kit that's proven itself in many of our live streams!
Matrox offer a comparable range of external Thunderbolt and internal PCIe cards that are great for capturing multiple HDMI or SDI inputs.
We've recently discovered Magewell's HDMI to USB 3.0 capture cards. They're relatively low cost, extremely compatible and seem amazingly simple to use. Thanks to Rasper for the tip!
HDMI vs SDI and Converters
When deciding between HDMI and SDI connectors for you cameras, it's important to note the key difference in the connectors (other than durability) is the fact that HDMI signal can only run a short distance (often less than 10m). Signal over SDI runs much longer.
If you use more affordable camera equipment that only supports HDMI, you may want to look at buying an HDMI to SDI converter to run cable over longer distances.
Other converters are also available.