What is an antenna chain and how do I figure out the correct setting for my router?

Short version: Using both radio chains is potentially twice as fast as just using one. You only want to limit yourself to a single radio chain if you're doing a long-distance outdoor link and trying to save money on antennas, or perhaps for certain troubleshooting situations.

Long version:
802.11n introduced the concept of MIMO (Multiple Ins, Multiple Outs) to Wi-Fi networking. MIMO radios use multiple radio chains working in parallel to transmit twice as much data at a time as a single radio could do.

You always want both chains in operation for the fastest rates. With just one chain, you're not doing MIMO, you just doing traditional SISO (Single In, Single Out), so you're not much better off than 802.11a or 802.11g. A and G used 20MHz wide channels and got a maximum PHY rate of 54Mbps. If you use those same 20MHz-wide channels with 802.11n and limit yourself to SISO (a single radio chain), your maximum PHY rate only goes up to 72.2Mbps. This increase is only because 802.11n added a couple new higher-rate modulation schemes than A/G had.

But if you turn on both radio chains so you're doing 2x2:2 MIMO, your max PHY rate (again, with 20MHz channels, for a fair comparison with A/G) doubles to 144.4Mbps.

And it's not just the max PHY rate that doubles. The second radio chain doubles your set available rates. For each of the eight 1-spatial-stream rates, there is a 2-spatial-stream rate that doubles it.

Of course, 802.11n also allows you to double your channel width to 40MHz. Doubling your channel width has the bonus of increased efficiency, so it actually slightly more than doubles your max PHY rate. So SISO 802.11n with 40MHz channels can do 150Mbps, and 2x2:2 MIMO 802.11n with 40MHz channels can do 300Mbps.

The only time you'd want to disable a radio chain is if you were trying to do a long-distance outdoor link, and didn't have an antenna rig that could provide enough separation between your two radio/antenna chains. If you didn't want to have to mount twice as many antennas on your antenna masts at each end of the link, and have them spaced far enough apart that they don't interfere with each other, then you might opt for a SISO link for cost/simplicity sake, even though it means only half the throughput (150 Mbps max PHY rate instead of 300).

I suppose one other reason to limit to a single radio chain would be temporary, for debugging things like interoperability problems. If you have a 1x1 802.11n device that isn't working well with your 2x2 AP, you might want to limit your AP to 1x1 temporarily to see if the 1x1 device works better. Maybe the 2x2 AP is buggy and sometimes tries to use 2x2 rates that the 1x1 device doesn't support, and ends up having to do increased retransmits or has increased packet loss.

Yes, "radio chain" is just a fancy way of referring to a transceiver. I don't know where they get the word "chain", but according to Motorola white paper: 802.11n Demystified (which was also referred by Steve (who asked the question) in a comment below Spiff's answer):

MIMO* introduces a new paradigm in RF systems design. MIMO-capable radios actually perform better within a multipath-rich environment. A MIMO system has multiple radio chains each of which is a transceiver with its own antenna. A radio chain refers to the hardware necessary for transmit/ receive signal processing. A MIMO radio can then apply several techniques to enhance signal quality and deliver more throughput. It is this ability to add signal components from multiple antennas that differentiates MIMO access points from traditional access points that use antennas in a diversity configuration. An access point with antenna diversity selects signal components from the antenna that provides the best signal performance and ignores the other antenna.

^{* MIMO = Multiple Ins, Multiple Outs}