These two blue-gloved men (agents of the Blue Sun Corporation? The Academy? the Alliance?) are pursuing River after her escape from The Academy. They always appear as a pair ("Two by Two"), have a generally creepifying demeanor, and wear blue gloves at all times.
The Hands of Blue were last seen in the short comic book series "Serenity: Those Left Behind", wherein it was revealed that their "gloves" did in fact extend under their clothing and were in actuality some kind of full body suit. It is evident that this revelation was not left over from the TV series as on the DVD commentary Joss Whedon himself admits that he just wanted them to wear the blue gloves and had no reason for doing so other than aesthetics. The pair also met their end towards the conclusion of the limited series when their ship was badly damaged by Serenity's full burn. The last few pages depict the passing of their mission onto the Operative whose superior refers to the Hands of Blue as private "contractors".
It is possible that the Hands Of Blue are an earlier-generation version of the... creatures that children like River were being modified to become. Alternately, they may be partially-failed experiments -- it's clear that it is only River's inherent genius that allows her to process all of the information that she is subject to, now that her mind has been modified -- in a less-apt subject, that input would be overwhelming and further mental modifications (very much like a lobotomy) would be necessary to prevent psychosis from setting in entirely. Certainly, the unusually calm, unemotional behavior of the Hands of Blue is reminiscent of the behavior of certain types of lobotomy subjects, or a sign of advanced anti-social personality disorder such as sociopathy, or the more notably lethal (as appropriate in this case) psychopathy.