I'm trying to work out the two possible values for B when A, a and b are all known. I'm certain its possible but I'm not sure how to start or what theories to look for to solve the question as my math is severely lacking. Any pointers on how to solve this problem or even just what to look for in order to do so would be greatly appreciated.
Note: Apologies for the vagueness, I'm a long time stack exchange user so I understand this question may have already been answered but I currently lack the terminology to try and search for the answers. If after understanding the solution better I find other answers on the site I will gladly remove this one.
For the calculation, I would suggest using the Sine Law. Call the angle at $C$ (this could be $C_1$ or $C_2$) by the name $C$, and call the angle at $A$ by the name $A$.
I will call the length that you called $b$ by the name $c$. I suggest you relabel your diagram. With that change, we have $$\frac{\sin(A)}{a}=\frac{\sin(C)}{c}.$$ You know $A$, $a$, and $c$. So from the above formula we have $$\sin(C)=\frac{c}{a}\sin(A).$$ Calculate. There are three possibilities.
Perhaps you will get $\sin(C)\gt 1$. This is impossible, so in that case there is no triangle satisfying the conditions.
Perhaps you will get $\sin(C)=1$. Then the angle at $C$ is a right angle, and there is a unique triangle that satisfies your conditions.
Perhaps you will get $\sin(C)\lt 1$. Then there are two possibilities for $\angle C$. One of them (call it $C_2$, to fit in with your diagram) is between $0$ and $90^\circ$. The other one is $180^\circ-C_2$. Call it $C_1$.
By using the fact that the angles of a triangle add up to $180^\circ$, you can now calculate the two possible angles $B_2$ and $B_1$.
Now you know all the angles, for each of the two possibilities. We can now use the Sine Law or the Cosine Law to find the remaining unknown side in each case.