Assume that you are given a straight line AB and a point C in a plane             
(Figure 1).   In Figure 1 the straight line AB is shown in black. 
You have to construct (to draw) the straight line parallel to AB 
and passing through the point C using a compass and a ruler. 

The construction is made in two steps. 

1. Using the ruler, draw an arbitrary straight line (AC in Figure 2) 
    passing through the given point C and cutting the given straight 
    line AB. 
    In Figure 2 the straight line AC is shown in the green color.
    Since the point A is an arbitrary point in the line AB, we 
    can denote as A the intersection point of the black and green lines.

  Figure 1. To the construction         

        Figure 2. To the solution
      of the construction problem

2. Using the ruler and the compass, construct the angle ACD with the vertex at the point C, adjacent to the straight line AC and congruent to the angle BAC
    The angle ACD should be located in the half-plane opposite to that where the angle BAC is located. 
    The way and the algorithm of constructing such an angle is described in the lesson How to draw a congruent segment and a congruent angle using a compass and a ruler 
    under the current topic Triangles of the section Geometry in this site. 

I am stating now that the side CD of the angle ACD is the straight line that have to be constructed. 

We need to prove that the straight line CD passes through the point C and is parallel to AB.
The straight line CD does pass through the point C by the construction, indeed.
The straight line CD is parallel to AB, because the alternate interior angles ACD and BAC are congruent by the construction (see the lesson Parallel lines under the topic 
Angles, complementary, supplementary angles of the section Geometry in this site).