Good folks!
If I may ask a relatively simple question when it comes to algebraic geometry... I was wondering if one of you people might help me in explaining in more detail an example that is in fact contained in the very opening sentence of Wikipedia's article on schemes:
The equations $x = 0$ and $x^2 = 0$ define the same algebraic variety and different schemes.
In as far as defining the same varieties, I find that to be pretty straightforward. A variety is merely the zero locus of a set of polynomial equations, and so, since $V(x)=V(x^2)$, they define the same variety.
I would very much appreciate it if someone could explain in as straightforward terms as possible how they define different schemes?
Even though there are nice answers in the comments, let me add a more philosophical answer.
As you said, varieties are just the zero sets of some polynomials and therefore cannot see multiplicities of the zeroes. But somehow the multiplicities should be taken into account as well and the notion of schemes does that. If there is a multiple zero, the point of the scheme corresponding to that zero can be visualized as being a "thicker" point and you can see that "thickness" as nilpotence in the local rings that come with a scheme. Schemes not having "thicker stuff" are called reduced and that notion is maybe also known to you for rings. A reduced ring is a ring not having non-zero nilpotent elements. As Ruben already mentioned, for your example it comes down to comparing $k[x]/(x) \cong k$ and $k[x]/(x^2)$. There you can explicitly see, that $x$ is nilpotent in the latter ring as $x^2 = 0$ in $k[x]/(x^2)$ and that is making the point thicker.