Why use the term 'models' to interpret the double turnstile symbol? A $\vDash $B can be read in words as: 


*

*A entails B

*B is a semantic consequence of A

*A models B


The first two are fine. But the third one seems a bit counter-intuitive to me. Somehow, I can't reconcile the term 'model' (used colloquially) with the notion of entailment or consequence.
Why do mathematicians and logicians use the term 'model' in such contexts? How is the term associated with the idea of entailment? And even more bizarre to me is the use of the $\vDash$ symbol in different contexts. For example, let A be an assignment, and F is an atomic formula. So that A $\vDash$ F (again, read as A 'models' F) means that A assigns a truth value of 1 to F. Again, this is counter-intuitive, and often had me wondering: "what is the underlying concept or idea behind it?"

The 'reading' #3 is a different thingy from reading #2 and #1.
As far as I know, the double turnstile ($\vDash$) has two uses.
In your reading #3, $A$ is some 'structure', which can be called a 'model', an interpretation, or an 'assignment', while $B$ is a propositional formula. The notation is more like $\mathcal{A}\vDash B$ or $\mathcal{M}\vDash B$, and it should be read/understood as "The structure $\mathcal{A}$ (or $\mathcal{M}$) models B" or "B is true in $\mathcal{A}$ (or $\mathcal{M}$)."
For example, in classical propositional logic, an assignment $\mathcal{A}$ is a two-valued function $A:B\rightarrow \{\top,\bot\}$ where $B$ is a propositional formula.
In readings #1 and #2, $A$ and $B$ are both propositional formulas. That is the 'semantic consequence' that you understand. The notation is now $A\vDash B$. Moreover, this means that for all assignments $\mathcal{A}$, if $\mathcal{A}\vDash A$, then $\mathcal{A}\vDash B$.
