There are LOTS of ways to construct parallel lines through a given point, because there are many properties of parallel lines that we can draw from (haha... get it? Draw from?). In this video, the person uses the property that when two parallel lines are cut by a transversal, corresponding angles are congruent. The converse of that is also true, so you can use that property to construct parallel lines. He draws two intersecting lines, labeling the vertex A, and then copies one of the angles. He then copies that angle (see other description) through point P and on the original line, which constructs parallel lines.
Draw a line segment, AB. Place your still point on A, and set your compass so that it measures the length of AB. Draw an arc. Without moving your compass settings, set it on point B, and swing the arc so that it intersects with the second arc. Draw a segment that connects A to the arc's intersection point, and another that connects B. This forms an equilateral triangle because all of the segments are congruent, because the vertices are the intersection points of congruent arcs.
Draw a segment that is less than double the furthest length your compass can reach. Label the endpoints A and B. Put your still point on A and adjust your compass so that it is wider than the halfway point of the segment. swing an arc. Without adjusting your compass, do the same process, starting at point B. Your arcs should intersect at two points that are equidistant from line segment AB and when connected, form a perpendicular bisector of the segment, separating it into two equal parts, and making a 90 degree angle from AB.
This video provides Instructions on how to copy an angle with a compass and a straightedge. First you make or find an angle that you wish to copy. You draw a segment elsewhere and label the vertex at the end. From that vertex, swing an arc, and also swing the arc on your original angle. Next, line your compass up where the arc intersects one of the segments, and put the still point on where it intersects the other segment. Fix that setting so that it stays put. Move your compass to the copied angle and put your still point where the arc intersects the segment. Swing another arc from there until it intersects with the first arc. Take your straightedge and connect the vertex with the intersection point of the two arcs. This should be your copied angle.