1NF: There should not be any row in a database table which has more than one value for any attribute in the table. The order of rows does not matter, neither does the order of columns but all the values of a cell must be part of the domain of the corresponding attribute.
2NF: The table should be in first normal form and there should not be any partial dependency in the table. A dependency A→B is partial if B is a non prime attribute and A is a prime attribute but not the candidate key of the table. An attribute is prime if it is part of any candidate key of the table, else it is a non prime attribute.
3NF: The table should be in second normal form and there should not be any transitive dependency. A dependency A→B is transitive if A and B are both non prime attributes i.e. neither A nor B is part of any candidate key.
BCNF: A table is in Boyce-coded normal form (BCNF) if for any dependency in the table A→B, A must be a super key of the table. 3NF allows dependencies in which left side of the dependency may be a non-prime attribute or a prime attribute but the right side is a prime attribute, which defeats the whole purpose of the candidate keys and prime attributes in general.