In lay terms, an FPGA is a chip that rewrites itself to be customized for a given use (or uses). Can rewrite themselves as many times as needed and very fast, too. Flexibility means more price, more watts, and less performance than truly custom chip. Yet, having custom circuits for the job can make an application SCREAM with performance. Here's some concepts for you.

1. Compression, encryption, etc that's many times faster (50x on some algorithms).

2. Do high-end streaming workloads (eg HD video, NIDS) on embedded hardware with hardly any watts.

3. Put specialized audio, AI, whatever engines in the FPGA for video games that take it to the next level with a whole CPU left for main game logic.

4. Implement a concurrent, hardware, garbage collector to write your whole OS in memory-safe language and not have freezes due to GC.

5. Use onboard I/O, often many lanes at Gbps, to get crazy throughput on any number of disk, networking, wireless, etc use-cases.

6. Use custom I/O for real-time applications.

7. Simulate other forms of hardware on the FPGA for personal learning or product development. Can deploy it in production on FPGA boards later.

8. Forget software emulators: build the hardware itself on the FPGA and have accurate simulation.

9. Cutting edge techniques do something similar with mockups of normal hardware which are modified to spot the exact time and place certain bugs happen. Then you can see everything from the input that caused it to the internal state of the processor. And fix it.

10. My use-case: processors modified to prevent code injection or data leaks to run applications that hackers can't hit. Altera's I'd use for prototyping then put them on anti-fuse FPGA's: write-once FPGA's that blow circuits intentionally to prevent attackers or glitches from modifying the system's logic. ROP that, bitches!

In short, you can use FPGA's for anything you can use a custom circuit for. They'll just be a bit weaker and usually depend on a host system to set them up. Common case is to have main CPU do most of the work with FPGA's accelerating it or handling interfaces (I/O) in a way they're better at. I'd say Google on FPGA hobby projects, use cases, "applications," etc to see the $3+ billion worth of uses for them. In case you're not drooling, here's a startup that's currently the top performer:

http://www.achronix.com/products/speedster22ihd.html

That joker has over a Gigahertz speed, 6 DDR3 lanes, up to 64 lanes of 12.75Gbps serial I/O, up to 16 of 28Gbps serial I/O, up to 400Gbps Ethernet, 400Gbps Interlaken (datacenter thing), and 2 PCI/Express controllers. Add 1.7 million LUTS for custom logic w/ 138 Mbits of registers or cache... noting that one of those mighty Oracle SPARC T1 processors only needs about 400,000 LUTS... to get a beast of a machine. Those are $10,000 unsurprisingly. Yet, even units that are several hundred dollars can do a hell of a lot and Altera on Intel's process node will do more.

I agree with another commenter that we'll see them in servers and datacenters first. The reason is that they'll have to charge more to recoup the initial cost of making them. Chips aint cheap: probably $5-10+ million per silicon test on Intel's node with mistakes requiring you to spend again. Tools to reduce that start at $1+ million a seat. Good thing FPGA's don't cost all that and have free/cheap synthesis tools. ;)