Really what it means is "Category 3" certified, with an additional review of a self-destruct situation to prevent breaking the nuclear payload.
Category 3 is also what is required to launch humans, and in fact the final milestone in the Commercial Crew is to rate the system Category 3. It is required for any kind of sensitive launch. Note that man rated requires a category 3 rocket, but also extra levels of scrutiny as well. A Category 3 launcher can carry cargo that is rated NPR 8705.4 categories A-D. Basically it means that NASA has done an extensive analysis of the rocket system, and/or had flight tests to minimize the risk. The requirements are quite extensive, and are documented at the link above. At the very least 3 consecutive flights are required to get the certification, and depending on the level of NASA review, that could be 14 consecutive flights. Note that one must also seek the certification, it's not automatically given upon the 14th consecutive launch of the rocket!
Today there are actually two US rockets that are certified Category 3 by NASA, the Atlas 5, and the Delta II. And of course, the Delta II is being phased out of service, in fact, there is only 1 left for sale. I've also seen a reference to the Delta IV being rated Category 3, which I'm assuming is only the Medium version, not the Heavy version.
In addition to being human rated, there are some additional requirements to being certified to launch a nuclear payload. The best reference I can find is this book, which I only have the Google edition with missing pages. The main difference seems to be an analysis of what would happen in the event of a Self-Destruct, specifically relating to high speed shrapnel heading through the payload, potentially damaging the casing protecting the nuclear material.