Sometimes this happens through relatively banal or logical reasons. For instance, murder has been labeled illegal and deviant by our society, and that makes good sense, as it would make our day-to-day lives significantly more difficult if people were murdering each other nilly willy.
But often such constructions are not neutral decisions arrived at after careful, logical consideration, but are rather the result of powerful interests attempting to (consciously or otherwise) entrench and/or expand their power, typically through enshrining their behavior as legitimate and the behaviors of others as illegitimate.
A wonderful example of how power influences social constructions can be seen in the current debates over the NSA leaks. As many spineless media commentators have pointed out in their attempts to bend over backwards providing justification for the most expansive spying program the world has even seen, nothing the NSA does/did is/was technically illegal (except for obviously directly contradicting the 4th amendment, but details, details). As the ever-on-point Glenn Greenwald explained, though, this is only the case because the very people doing this also wrote the laws governing it.
This is why the NSA is such a good example of constructionism -- Edward Snowden is currently an international fugitive for doing a much, much smaller version of what the NSA is doing. It is literally the exact same behavior, only done on a much smaller scale and arguably for much more noble reasons. For the man without power, this small-scale internet surveillance means he will likely spend the rest of his life in prison, while the people with power are allowed to continue doing that same action on a much larger scale and not only not be considered criminals, but instead have the brazen gall to call themselves security experts.
Of course, this doesn't even get in to the fact that much of the the NSA is doing actually is illegal, and that the technically illegal actions of Snowden were necessitated by the fact that there is literally no other way for this information to have been made public. And that's not an abuse of the word literally; if you read the linked article, you'll learn that multiple groups have sued the US government over the NSA, often simply to learn the extent of information gathered, not even what the actual information was. And yet they can't do that because:
In dismissing the case, the court agreed with the precedent set in two other cases, which basically said that Americans don’t even have the right to sue their government over its surveillance program, unless they can prove that their communications were intercepted. Of course, that’s essentially impossible since the program is classified and you can’t use classified documents in court, even if you somehow got your hands on them.
So to recap: leaking classified documents proving the highest levels of government have been routinely violating the 4th amendment in ways which violate both domestic and international law with no probable cause for the vast majority of their actions, and you're a terrorist who must like on the run. Set up and participate in the program which violates both domestic and international law, and you're a beacon of freedom, defending our nation from terrorists.
That's pretty much a textbook example of the social construction of deviance.