Implicitness can refer to a few different concepts. Broadly, something implicit is a trait that is understood, or implied, even though it is not directly expressed. As an example, there was an implicit understanding within a family that no one discussed their relative's substance abuse. Additionally, implcit memories (procedural memories) such as riding a bike, driving a car, etc. once learned become nearly automatic skills that require little thought.
In social and cognitive psychology, as humans we acquire implicit attitudes, feelings, and beliefs that are learned from familial upbringings and social environments without anyone explicitly expressing these ideas. Implicit attitudes can include prejudices, stereotypical thinking, and particular ways of reacting to circumstances. Implicit attitudes are similar in that they are attitudes that have become so ingrained and automatic that they exist with little or no thought being given to how they developed or if they are fair or legitimate beliefs - common examples include prejudice and bigotry.
The IAT, or Implicit Association Test, is a test that exists in social psychology to measure implicit attitudes - the power of a person's automatic associations between mental representations of objects concepts in memory. Simply put, this test evaluates and exposes deeply ingrained attitudes and prejudices. There is some controversy regarding the IAT in that it is isn't truly certain if it measures anything beyond the existence of these implicit memories.