Parabolas are described in terms of their concavity. Concavity refers to the word "concave," which means shaped similar to a hole or inner surface of a sphere. When mathematicians speak of a specific parabola, they like to describe it as being "concave up" or "concave down." These terms describe the "direction" of a parabola. This direction lets mathematicians know whether the parabola has a maximum or minimum value (it can only have one).
The tangent of a function is the line touching the function at only a single point. The tangents of parabolas are special in that they always have slopes that are linear functions of the "x" variable. This is because parabolas are written as a function with an "x^2" term. When using calculus, you will find that the derivative of a parabola always contains a "cx" term as its highest term, with "c" representing a constant number. The implication is that the slope of the tangent at any point of a parabola is linear.
All parabolas have a single vertex. The vertex is the point where the parabola seems to originate from. It is also the point at which the tangent's slope is equal to zero. If you were to draw a vertical line through the vertex of a parabola, you would be drawing the line of symmetry; the right and left sides of the parabola would be equal in appearance.
Parabolas, being conic sections (dissections of a cone), have unique focal points. These focal points do not exist on the parabola but instead in the concave area vertically out from the vertex. The focal point of a parabola has many important applications. The focal points of parabolas are used in designing microscopes and telescopes as well as in predicting the path of satellites and asteroids.