As an exammple of the scheme, let's try the circle of radius $\sqrt{5}$ which is the equation $X = \{ x^2 + y^2 = 5 \}$. This circle has four integer points $(\pm 2, \pm 1)$ and $(\pm 1, \pm 2)$.
As a Scheme, I merely have to define the coordinate plane here over integers $\text{Spec}\,\mathbb{Z}[x,y]$ - which has the integer points $\mathfrak{p} = (x-a,y-b)$ with $a,b \in \mathbb{Z}$ (among others). The circle is then: $$ X = \text{Spec}\mathbb{Z}[x,y]/(x^2 + y^2 - 5) $$ This scheme contains information about the behavior of a circle modular arithmetic
- $p= 2$ then $x^2 + y ^2 - 5 = (x-y+1)^2 $ is double-line
- $p= 3$ then $x^2 + y ^2 - 5$ is a circle
- $p= 5$ then $x^2 + y ^2 - 5 = x^2 - 4y^2 = (x+2y)(x-2y)$ is two lines
- $p= 7$ then $x^2 + y ^2 - 5$ is a circle
What is the tangent space to this scheme at $(x,y) = (2,1)$. This would correspond to the ideal $(x-2,y-1) \subseteq \mathbb{Z}[x,y]$ modulo the circle $(x^2 +y^2 - 5)$. My best guess is that these should be the integer points of the tangent line $$\ell = \{ 2\times (x-2)+1\times (y-1) = 0 \} \cap \mathbb{Z}$$ Perhaps the Zariski tangent space will have other information.
The definition of the tangent space of a Scheme is rather technical $$ T_x^*X = \mathfrak{m}_x/\mathfrak{m}^2_x \text{ with }\mathfrak{m}_x = (x-2,y-1)\Big[ \mathbb{Z}[x,y]/(x^2 + y^2 - 5) \Big]_{(x-2,y-1)} $$ What might this localization be? And the definition of the tangent space requires computing the perfect squares.