Ineffable cardinal
Ineffable cardinals were introduced by Jensen and Kunen in [1] and arose out of their study of $\diamondsuit$ principles. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ is stationary. Equivalently an uncountable regular $\kappa$ is ineffable if and only if for every function $F:[\kappa]^2\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^2$ is constant [1]. This second characterization strengthens a characterization of weakly compact cardinals which requires that there exist such an $H$ of size $\kappa$.
If $\kappa$ is ineffable, then $\diamondsuit_\kappa$ holds and there cannot be a slim $\kappa$Kurepa tree [1] . A $\kappa$Kurepa tree is a tree of height $\kappa$ having levels of size less than $\kappa$ and at least $\kappa^+$many branches. A $\kappa$Kurepa tree is slim if every infinite level $\alpha$ has size at most $\alpha$.
Contents 
Ineffable cardinals and the constructible universe
Ineffable cardinals are downward absolute to $L$. In $L$, an inaccessible cardinal $\kappa$ is ineffable if and only if there are no slim $\kappa$Kurepa trees. Thus, for inaccessible cardinals, in $L$, ineffability is completely characterized using slim Kurepa trees. [1]
If $0^\sharp$ exists, then every Silver indiscernible is ineffable in $L$. [2]
Relations with other large cardinals
 Measurable cardinals are ineffable and stationary limits of ineffable cardinals.
 $\omega$Erdős cardinals are stationary limits of ineffable cardinals, but not ineffable since they are $\Pi_1^1$describable. [2]
 Ineffable cardinals are $\Pi^1_2$indescribable [1].
 Ineffable cardinals are limits of totally indescribable cardinals. [1] ([3] for proof)
Weakly ineffable cardinal
Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in [1] as a weakening of ineffable cardinals. An uncountable regular cardinal $\kappa$ is weakly ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ has size $\kappa$. If $\kappa$ is weakly ineffable, then $\diamondsuit_\kappa$ holds.
 Weakly ineffable cardinals are downward absolute to $L$. [1]
 Weakly ineffable cardinals are $\Pi_1^1$indescribable. [1]
 Ineffable cardinals are limits of weakly ineffable cardinals.
 Weakly ineffable cardinals are limits of totally indescribable cardinals. [1] ([3] for proof)
Subtle cardinal
Subtle cardinals were introduced by Jensen and Kunen in [1] as a weakening of weakly ineffable cardinals. A uncountable regular cardinal $\kappa$ is subtle if for every for every $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ and every closed unbounded $C\subseteq\kappa$ there are $\alpha<\beta$ in $C$ such that $A_\beta\cap\alpha=A_\alpha$. If $\kappa$ is subtle, then $\diamondsuit_\kappa$ holds.
 Subtle cardinals are downward absolute to $L$. [1]
 Weakly ineffable cardinals are limits of subtle cardinals. [1]
 Subtle cardinals are limits of totally indescribable cardinals. [1]
 The least subtle cardinal is not weakly compact as it is $\Pi_1^1$describable.
 $\alpha$Erdős cardinals are subtle. [1]
Completely ineffable cardinal
Completely ineffable cardinals were introduced in [3] as a strenthening of ineffable cardinals. Define that a collection $R\subseteq P(\kappa)$ is a stationary class if
 $R\neq\emptyset$,
 for all $A\in R$, $A$ is stationary in $\kappa$,
 if $A\in R$ and $B\supseteq A$, then $B\in R$.
A cardinal $\kappa$ is completely ineffable if there is a stationary class $R$ such that for every $A\in R$ and $F:[A]^2\to 2$, there is $H\in R$ such that $F\upharpoonright [H]^2$ is constant.
 Completely ineffable cardinals are downward absolute to $L$. [3]
 Completely ineffable cardinals are limits of ineffable cardinals. [3]
$n$ineffable cardinal
The $n$ineffable cardinals for $2\leq n<\omega$ were introduced by Baumgartner in [4] as a strengthening of ineffable cardinals. A cardinal is $n$ineffable if for every function $F:[\kappa]^2\rightarrow n$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^n$ is constant. A cardinal $\kappa$ is totally ineffable if it is $n$ineffable for every $n$.
 $2$ineffable cardinals are exactly the ineffable cardinals.
 an $n+1$ineffable cardinal is a stationary limit of $n$ineffable cardinals. [4]
 a $1$iterable cardinal is a stationary limit of totally ineffable cardinals. (this follows from material in [5])
References
 Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www bibtex

Jech, Thomas J. Set Theory. Third, SpringerVerlag, Berlin, 2003. (The third millennium edition, revised and expanded) bibtex

Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and Zwicker, William. Flipping properties: a unifying thread in the theory of large cardinals. Ann Math Logic 12(1):2558, 1977. MR bibtex

Baumgartner, James. Ineffability properties of cardinals. I. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I, pp. 109130. Colloq. Math. Soc. János Bolyai, Vol. 10, Amsterdam, 1975. MR bibtex

Gitman, Victoria. Ramseylike cardinals. The Journal of Symbolic Logic 76(2):519540, 2011. www arχiv MR bibtex