The law of sines provides a formula that relates the sides with the angles of a triangle. This formula allows you to relatively easily find the side length or the angle of any triangle.
Law of Sines Formula
When to use the law of sines formula
You should use the law of sines when you know 2 sides and an angle (case 1 in the picture below) and you want to find the measure of an angle opposite a known side. Or when you know 2 angles and 1 side and want to get the side opposite a known angle (case 2 in picture below). In both cases, you must already know a side and an angle that are opposite of each other.
Cases when you can not use the Law of Sines
The picture below illustrates a case not suited for the law of sines. Since we do not know an opposite side and angle, we cannot employ the law of sines formula.
By the way, we could use the law of cosines to find the length of the side opposite the 115° angle. Once we know that side length ,we'd be able to use the law of sines as indicated by case 1 in the picture up above.
Video
on how to use the Law of Sines
Practice Problems
set up ratio with known pair of side/angles
$$ \frac{ sin( \color{red}{ b})}{ 16} = \frac{ sin(115)} {123} $$
solve for unknown
$$ \frac{ sin( \color{red}{ b })}{ 16} = \frac{ sin(115)} {123} \\ sin( \color{red}{ b }) = \frac{ 16 \cdot sin(115)} {123} \\ sin( \color{red}{ b }) = 0.11789369587468619 \\ \color{red}{ b } = sin^{1} ( 0.11789369587468619 ) \\ \color{red}{ b } = 6.770557323410266 \\ \color{red}{ b } \approx 6.8 $$
Yes, first you must remember that the sum of the interior angles of a triangle is 180 in order to calculate the measure of the angle opposite of the side of length 19.
Now that we have the measure of that angle, use the law of sines to find value of x
solve for unknown
$$ \frac { \color{red}{x} }{ sin(116)} = \frac{19}{sin (34) } $$
then cross multiply to calculate x.
x = 30.5
When there are no Triangles!
So if you are given any 2 sides and an angle, can you always use the law of sines to find all the other sides and angles of a triangle?
The answer is Not always!
Consider the following. Let's imagine that we know that there is some 'triangle' ABC with the following information:
BC = 23 ,AC = 3, = 44.
Below represents an accurate picture of these sides and angles. Red represents a given of the triangle (2 sides and 1 angle). Grey represents a variable that we can change (the grey 'side' AB). The gray line represents the third side of the triangle that we don't know. The only thing we know about that side is that it must extend at 44 degrees. As you can see from the picture, there is no way for that grey side to rise at 44 degrees and also to intersect with point A. Therefore no triangle can be drawn with these givens.
Impossible 'Triangle'
Side BC = 23
Side AC = 3
$$ \angle B$$= 44
Let's see what happens, if we try to use the law of sines to determine the length of "side AB,", the grey line in the picture on the left.
You know the deal, lets use our law of sines formula
$$ \frac{sin(44) }{3 } = \frac{sin(a)}{23} \\ sin(a) = \frac{23 \cdot sin(44)}{3} = 5.32571417 \\ a = sin^{1}( \color{red}{ 5.32571417}) $$
As you can see angle A is 'impossible' because the sine of an angle cannot be equal to 5.3. (Remember the greatest value that the sine of an angle can have is 1)
Double triangle Problems
Use the fact the sum of the interior angles of a triangle is 180° to calculate all of the angles inside the triangles.
Now, use the law of sines formula to set up an equation.
From here you have several options
The easiest one is to use sohcahtoa to solve for a side length to arrive at the answer.

Further Reading:
 Ambiguous Case of the Law of Sines
 law of cosines vs law of sines
 When to use the law of sines and when to use law of cosines!
 Law of Cosines
 law of sines worksheet(includes answer key, model problems and visual aides)
 Triangle Calculator (calculates unknown sides/angles using Law of Sines, can tell you have how many triangles can be created and more)