TITLE: How to tell if algebraic set is a variety?
QUESTION [10 upvotes]: I've been reading some basic classical algebraic geometry, and some authors choose to define the more general algebraic sets as the locus of points in affine/projective space satisfying a finite collection of polynomials $f_1, \dots, f_m$ in $n$ variables without any more restrictions. Then they define an algebraic variety as an algebraic set where $(f_1, \dots, f_m)$ is a prime ideal in $k[x_1, \dots, x_n]$. 
My question has two parts: 

I'm guessing the distinction is like any other area of math where you try to break things up into the "irreducible" case and deduce the general case from patching those together. How does that happen with varieties and algebraic sets? Is it correct to conclude that every algebraic set is somehow built from algebraic varieties since the ideal $(f_1, \dots, f_m)$ is contained in some prime (maximal) ideal? 
How can one tell whether or not an algebraic set is a variety intuitively? I know formally you'd have to prove $(f_1, \dots, f_m)$ is prime (or perhaps there are some useful theorems out there?), but many times in texts the author simply states something is a variety without any justification. Is there a way to sort of "eye-ball" varieties in the sense that there are tell-tale signs of algebraic sets which are not varieties? 

Perhaps this is all a moot discussion since modern algebraic geometry is done with schemes and this is perhaps a petty discussion in light of that, but nonetheless, I'd like to understand the foundations before pursuing that.
Thanks.

REPLY [3 votes]: Here are some personal reflections on the role of schemes in algebraic geometry, commenting on your very interesting remark: "Perhaps this is all a moot discussion since modern algebraic geometry is done with schemes".   
Grothendieck introduced scheme theory in the late nineteen fifties and the high level of abstraction of that theory had a discouraging effect on even great mathematicians like Néron.
Soon however the incredible power of these new tools allowed people like Deligne Grothendieck and Faltings to solve problems untouchable by classical methods: Grothendieck and Deligne solved all of the Weil conjectures and Faltings solved conjectures of Mordell, of Shafarevich and of Tate.
Fields medals were of course awarded to Grothendieck, Deligne and Faltings and several other medals were  won by  mathematicians whose work involved in a fundamental way  scheme theory: Lafforgue, Mumford, Ngô, Voevodski,...  
For us lesser mortals scheme theory has now become quite accessible thanks to the didactic efforts of pioniers like Mumford and Hartshorne and then thanks to the excellent  textbooks by Eisenbud-Harris, Liu, Görtz-Wedhorn,... and on-line notes like Vakil's splendid Foundations of Algebraic Geometry.  
Algebraic varieties retain an enormous place in contemporary algebraic geometry.
However an overwhelming part of the results in that field are obtained with the help of scheme theory.
So every algebraic geometer must at some time learn scheme theory, but the best way to approach algebraic geometry is through one of the many introductory books written in the classical language by experts (who  of course also master scheme theory!) like  Fulton, Hulek, Perrin, Reid, ...