Some terminology ...
A frame is a pair $\mathfrak F = \langle W, R \rangle$, where $W$ is a non-empty set (the set of "possible worlds") and $R$ is a partial order on $W$.
A valuation $V$ is a map associating to each sentential variable $p$ of the language a subset $V(p) \subseteq W$.
A model is a pair $\mathfrak M = \langle \mathfrak F, V \rangle$.
We say that a sentential variable $p$ is true at $w$ if $w \in V(p)$.
Let $\mathfrak M = \langle \mathfrak F, V \rangle$ a model and $x$ a point in the frame $\mathfrak F = \langle W, R \rangle$.
We define, via the usual inductive definition on the complexity of $\varphi$, the relation :
"$\varphi$ is true at $x$ in $\mathfrak M$"
and we write :
$(\mathfrak M, x) \vDash \varphi$,
starting from : $(\mathfrak M, x) \vDash p$ iff $x \in V(p)$ [here we need $V$].
A formula $\varphi$ is true at a point $x$ in $\mathfrak F$ (notation : $(\mathfrak F, x) \vDash \varphi$) if $\varphi$ is true at $x$ in every model based on $\mathfrak F$.
$\varphi$ is valid in a frame $\mathfrak F$ (notation : $\mathfrak F \vDash \varphi$) if $\varphi$ is true in all models based on $\mathfrak F$.