In more detail
m = (y₂ − y₁) / (x₂ − x₁). Between (1, 2) and (4, 6) the slope is 4/3 ≈ 1.333 and the distance is 5. A positive slope rises to the right, a negative one falls, a horizontal line has slope 0 and a vertical line has no defined slope (division by zero). With the slope and one point you get the equation y = mx + b. Parallel lines share a slope; perpendicular slopes multiply to −1, so the perpendicular to 4/3 is −3/4.
Why it matters
Slope is the rate of change in any linear relationship: price per unit, speed on a distance-time graph, the trend of a Linear regression line. Roofs and stairs use rise over run ("7 in 12"), and roads use percent grade, 100 × rise / run. A slope of 1 is a 45° angle, so a 100% grade is 45°, not vertical. The angle is atan(slope).
Example
line through (1, 2) and (4, 6)
m = (6 − 2) / (4 − 1) = 4/3
b = 2 − 4/3 × 1 = 2/3 → y = 4/3 x + 2/3
8% road grade atan(0.08) = 4.57°
roof pitch 7 in 12 atan(7/12) = 30.26°
roof pitch 4 in 12 atan(4/12) = 18.43°
Common confusion
A straight line has one slope everywhere; a curve's slope changes from point to point and is given by its Derivative.