Social Desirability Bias
TLDR; Public opinions are often slanted in favor of the least controversial option.
This bias may also be known as the Shy Tory Factor, Bradley Effect, Wilder Effect, or Courtesy Bias. The gist of it is that we have a tendency to give an opinion publicly that is more socially correct than our privately held opinions.
Examples of Social Desirability Bias
Shy Tory Factor
The Shy Tory Effect was coined by British opinion polling companies in the early 1990s when election studies showed that the electoral vote won by the conservative party ("Tories") was far greater than the than the opinion polls would have predicted.
Tom Bradley Effect
Tom Bradley was a black mayor of Los Angeles serving from 1973 to 1983. In the 1982 Governor's race, Tom was ahead in the voter polls going into the election. Despite that lead in the polls, he still lost. The proposed explanation is that white voters would give innaccurate polling responses out of a fear of opening themselves up to racial criticism.
Accounting for Social Desirability Bias
Pollsters already try to account for this effect in major elections, but if you're conducting your own survey you might want to consider this bias and how it might slant your results.
One thing you can do is try to minimize the "shyness" factor by doing whatever you can to make participants feel secure in sharing their honest opinions. Making the survey anonymous is one way of of accomplishing that. Sometimes when a survey is anonymous, but digital, people will still not trust the system to be anonymous. You might consider using a different method.
You should also consider the level of controversy present in the survey. For example, if the results of the survey will result in crucial decisions that could greatly impact the people involved, you have a greater chance of this bias being a factor.