Strictly speaking, a tangent is a line drawn so that it intersects one point on a circle.
It is a line perpendicular to the radius at that point.

When you draw a tangent on a curve, just try to imagine where the radius would be if you had the whole circle.  Then add a line at 90 degrees to it.

I hope this picture helps.
 
OK, start out with some curvy shape
(1) Pick the spot where you want the slope
(2) Imagine a circle on the inside of the curve
(3) Where would the radius be that hits your point?
(4) Add your line at 90 degrees; this is your TANGENT and you just get its slope to find speed at original point!