A tangent line to a curve at a given point is the straight line that “just touches” the curve at that point. Perhaps the name “touching line” would be much better. As this line passes through the point where it meets the curve (point of tangency) this line is suppose to “going in the same direction” as the curve. Stated another way, a tangent line to a curve at a given point is the best straight-line approximation of the curve near that point.
This definition is probably very confusing. That’s because we didn’t use calculus. In the demo below we have a curve and two points (red) on the curve. We also have a line that cut through the curve at these two points. We call such a line a secant line of the curve (and perhaps “cutting line” is a much better name). Try use your mouse to move one of the point. As you move this point to the other point, the “cutting line” becomes the “touching line”. That is, the secant line becomes the tangent line. By keeping track of how the slope of the secant line changes as the two points move closer and closer together, we can actually “see” what the slope of the tangent line should be. (Can you?)