Sorry for the terse title; the 150-character limit really gets in the way when you have to include "\varepsilon" in the title several times.
Write $k[\varepsilon]$ for the dual numbers, i.e. $k[x]/(x^2)$.
If we glue two copies of $\operatorname{Spec} k[\varepsilon]$ along their closed points, we get $\operatorname{Spec} k[\varepsilon_1, \varepsilon_2]$.
In geometric terms, if we take two points, each with "one dimension's worth of tangent information" attached, and glue the points together, we get a point with "two dimensions' worth of tangent information attached." This is related to the fact that tangent spaces always really are vector spaces, even in cases where the tangent cone is not.
In contrast, if we glue, say, two copies of $\operatorname{Spec} k[x]$ together at one of their points we'd get $\operatorname{Spec} k[x,y]/(xy)$, a space where you can move away from the glue point in two distinct directions.
So if you glue two lines together you somehow preserve the fact that some directions are along the lines and others aren't, whereas if you glue two "infinitesimal lines" together you can no longer see what direction the infinitesimal lines had pointed in from the resulting space alone.
By "directions" here I mean that the inclusion of e.g. $\operatorname{Spec} k[x] \hookrightarrow \operatorname{Spec} k[x,y]/(xy)$ determines a tangent vector in $T_{(0,0)} \operatorname{Spec} k[x,y]/(xy)$.
I can calculate that all these things are true, but I've never really been able to internalize the idea. Does anyone have a nice way of thinking about this?