Let $V$ be a subset of $k^n$ given parametrically as $x_1 =g_1(t_1,...,t_m) ...x_n=g_n(t_1,...,t_m)$. If the $g_i$ are polynomials (or rational functions) in the variables $t_j$, then $V$ will be an affine variety or part of one. find a system of polynomial equations (in the $x_i$) that define the variety.
-ch. 2 of Cox, Little, O'Shea 's Ideals varieties and Algorithms
If I understand the definition of Variety any set of points is "an affine variety or part of one" but what is special about this...an example where this construction is used would be helpful (the simpler the better)
Yeah, this question is confusing if you're not already familiar with the subject. You're supposed to infer a particular meaning of "part of" that's obvious if you already know this stuff and meaningless if not.
The point is that the image of a morphism of varieties is a constructible set, and in particular contains a dense open subset of its closure. The variety that it's "part of" is the closure $\overline{V}$ of $V$, or equivalently the variety cut out by the ideal of all polynomials vanishing on $V$. The goal is to find a finite list of generators for this ideal.
To give a standard illustration, consider
$$x_1 = t_1, \qquad x_2 = t_1 t_2, \qquad x_3 = 0$$
In this case, $V$ is the set of points $(p, q, 0)$ for $p \neq 0$, together with the point $(0,0,0)$. The set $V$ itself isn't very nice -- it's neither open nor closed nor even open in its closure. It's a plane, minus a line, with a point on that line added back in. The closure $\overline{V}$ of $V$, on the other hand, is simply the plane $x_3 = 0$, and the solution to the implicitization problem is therefore just the system $\{x_3 = 0\}$.