That right is a US constitutional guarantee, and is famous because it is mentioned as the first item of the "Miranda" list, which for several decades now has ritually been administered to arrestees for criminal offenses at an early stage in the process. As a consequence, it's very familiar to anyone who watches US-based police TV shows. It means that accused persons in the US have no obligation to tell police anything. Interrogations are supposed to stop when the accused requests an attorney.
In the bad old days, there was no legal protection against self-incrimination; basically, you could be beaten into admissions of guilt which could then be used to convict you. (The ultimate, perhaps, were the infamous courts of the various Inquisitions, which systematically tortured people in very large numbers in order to secure confessions which supposedly would save souls.) The framers of the Constitution rejected this obviously unjust practice, and the courts have tended to strengthen protections over the years.
Though many nations have similar protections--for instance, Canada has the Charter of Rights--some jurisdictions may have different standards, and protection against self-incrimination may be weak or even non-existent. Also, some countries or localities may not observe their own standards--and when crime rates are high and people are worried about their security, legal standards can look like foolish "technicalities" which only stand in the way of getting dangerous criminals off the street.
Then there is political pressure to erode protections. The risk, of course, is that the wrongfully accused then suffer wrongful conviction as a result.