When I study maths, I try to understand why the mathematicians brought out this concept or what usefulness they might have seen in the concept that they worked upon. So when I started with straight lines, what hit me was that why do we have to have so many forms of equations of straight lines? All they do is just help make more questions but don't really help us in real life applications of straight line concepts (or do they?). What was the need to explicitly define those forms? Why not only one general form was defined?
This started as a comment on an over-general and unspecific question, but got too long.
There are forms of equation which represent the fact that a line is defined by any two points on it, or by a single point and a direction. The point and direction form is useful in determining the tangent or normal to a differentiable curve.
Some forms distinguish between dependent and independent variables, others treat the variables as equivalent to each other. The vector form of a straight line generalises to multiple dimensions, and shows that the straightness of the line is preserved by a variety of transformations which preserve the essential vector structure. Some forms relate more naturally than others to the fact that the straight line is a geodesic in Euclidean space.
Sometimes it is useful to treat lines in projective co-ordinates, which are fully homogeneous. If there is a geometric structure in view, but no natural rectilinear co-ordinates, barycentric co-ordinates can come in handy.
Euclid did geometry involving straight lines without our notion of co-ordinates or equations - and the first form I mentioned - the line through two points, is a co-ordinate form of what Euclid meant by a line - and so relates co-ordinate geometry to Euclid's propositional geometry.