The data you are trying to insert already exists and its a primary key with unique value

 unique constraint (DATABASENAME.PK_COLUMN_NAME) violated

When you try to insert some value which are not allowed for the column this error is thrown.Similar to ENUM of Java.

In the below example CATEGORY Column may have MALE and FEMALE But when you try to insert BOTH this is going to throw error since it is not allowed value.

CATEGORY IN ('MALE','FEMALE')

check constraint (DATABASENAME.CHK_CATEGORY) violated

If the Parent table referred by Foreign Key does not have the value then this is thrown

integrity constraint (DATABASENAME.FK_CATEGORY) violated - parent key not found