People use "robustness testing" in very different ways, from what I have seen. I have mainly seen it used as a way to demonstrate effective exception handling within applications. (You will often hear this in the context of API testing, for example.) This is often tied in with a type of performance testing, such as the ability to actually provide 24x7 uptime in a Web farm environment, for example. In terms of things like hardware testing or environmental testing, it means just what it sounds like: testing how robust something is.
The clearest definition I have seen of it (and this ties in with other testing types) is: "the testing that attempts to cause failures involving how the system behaves under invalid conditions (e.g., unavailablility of dependent applications, hardware failure, and invalid input such as entry of more than the maximum amount of data in a field)." That may seem like stress testing. When people use this term this way, stress testing is defined as: "testing that attempts to cause failures involving how the system behaves under extreme but valid conditions (e.g., extreme utilization, insufficient memory inadequate hardware, and dependency on over-utilized shared resources)." So notice the different emphasis.
With this, you have to realize that "robustness" is defined as two things: (1) a quality requirement specifying the degree to which an application continues to function properly under abnormal circumstances. (2) a quantitative quality factor measuring the actual degree to which an application continues to properly function under abnormal circumstances. Some people will say that robustness includes the handling of things like failover, degraded modes of operation, disaster recovery schemes, etc. (You can also compare this with the notion of what people call operational availability or just availability, for short. You might see these two conflated in some Web environments or if you ever work for or with an Internet Service Provider.)
[This message has been edited by Cryptonomic (edited 12-26-2002).]