Question: Well, the question is; "Write a C code that finds zero of a function y = ax + b, without solving the equation. The zero will be found within the precision limit ϵ = 10 - 3. You'll start at x=0, and move x in the proper direction until |y|< ϵ."

I'm a newbie, to programming, and don't know anything about this ϵ thing either.
Help me out!!

Answer: 
It means you have to solve the inequality |ax+b| < 10^-3 by trying different values for x.

Since this is a linear function it's easy. Start with a random number at x and then increase it or decrease it depending on the result of ax+b. I.e. if you move to one direction and the results go more away then you should follow the opposite direction.

You will have to develop an algorithm that decides the increments/decrements of x.
|y| < 10⁻³, or well, -0.001 < y < 0.001.

You must increase or decrease x (starting from 0, as you've said) in order to make y to take a value between -0.001 and 0.001.

About ϵ, a.k.a. epsilon, is used to denote a very small value. For this problem, ϵ denotes a tolerance value, as it's not needed y to take a strict value of 0.