Definition: A median of a triangle is a segment from a vertex to the midpoint of the opposite side; every triangle has three medians.
What is a median?
Choose a vertex. Find the midpoint of the side across from it. The segment joining them is a median. It splits the triangle into two smaller triangles of equal area, since they share the same height and have equal bases.
How does it differ from other segments?
An altitude is perpendicular to the opposite side. An angle bisector splits the angle. A median goes to the midpoint. In an isosceles triangle, the median from the apex does all three jobs at once.
Example. Triangle ABC has A(0, 0), B(10, 2), C(4, 8). The midpoint of BC is ((10 + 4)/2, (2 + 8)/2) = (7, 5). The median from A has length the square root of 49 + 25, which is the square root of 74, about 8.6. The slope of the median is 5/7.
What mistakes occur?
Confusing the median with an altitude, and so drawing a perpendicular. Also, forgetting that the midpoint comes from the opposite side, not an adjacent side. The three medians meet at the centroid, which divides each in a 2 to 1 ratio.