1NF: The table faithfully represents a relation and has no repeating groups.
2NF: Non-prime attributes are non functionally dependent on a proper subset of a candidate key.
3NF: Non-prime attributes are non-transitively dependent on every candidate key in the table.
BCNF: Every non-trivial functional dependency is a dependency on a superkey.
4NF: Every non-trivial multivalued dependency in the table is a dependency on a superkey.
5NF: Every non-trivial join dependency in the table is implied by the superkeys of the table.
DKNF: Every constraint in the table is a logical consequence of the table's domain and key constraints.
Please let me know which, if any, of these requires further explanation. Each provides a means of checking whether a table is in the normal form.
EDIT: Again from Wikipedia:
A join dependency is a constraint on the set of legal relations over a database scheme. A table $T$ is subject to a join dependency if $T$ can always be recreated by joining multiple tables each having a subset of the attributes of $T$. If one of the tables in the join has all the attributes of the table $T$, the join dependency is called trivial.
The join dependency plays an important role in the Fifth normal form, also known as project-join normal form, because it can be proven that if you decompose a scheme $R$ in tables $R_1$ to $R_n$, the decomposition will be a lossless-join decomposition if you restrict the legal relations on $R$ to a join dependency on $R$ called $*(R_1,R_2,...R_n)$.
In other words, if a constraint on your table $T$ is that
T = R JOIN S for some tables $R$ and $S$, then $T$ has a join dependency $*(R, S)$.