How are the intersections and quotients of algebraic groups defined? Let $k$ be a field and we define $Spec(A)$ to be the functor from $Alg_k$ to $Sets$ such that it sends 
$$
R \rightarrow Hom_{Alg_k}(A, R). 
$$
We let functors of this form where $A$ is a $k$-algebra, an affine scheme. And then we let algebraic groups be an affine group scheme of finite type over $k$.   
This is a setting I'm not very used to, and I was wondering if someone could explain me (or provide me with some idea how it works)  
1) How does one take intersections of two subgroups of $G$ (algebraic group) in this setting? 
2) How does one consider the quotient $G/N$ when $N$ is a normal subgroup of $G$? 
Thank you. 
 A: This is a pretty complicated topic, so let me just give vague ideas and then give you references.
1) As Max pointed out in the comments above, if $H_1,H_2$ are closed algebraic subgroups of $G$ then $H_1\cap H_2$ has the functor which takes any $k$-algebra $R$ and associates to it $(H_1\cap H_2)(R)=H_1(R)\cap H_2(R)$. The reason that this is representable is that it agrees with the functor of points of $H_1\times_G H_2$. Indeed, note that since $H_2\hookrightarrow G$ is a closed embedding we know that the projection $H_1\times_G H_2\hookrightarrow H_1$ is a closed embedding. But, what is the points of $(H_1\times_G H_2)(R)$? It's the set $\{(a,b)\in H_1(R)\times H_2(R):a=b\in G(R)\}$ thus the inclusion $H_1\times_G H_2(R)\to H_1(R)$ evidently has image $H_1(R)\cap H_2(R)$.
Note though that $H_1\cap H_2$ needn't just be the set theoretic intersection with reduced structure. This is true in characteristic $0$, but in characteristic $p$ you note that things like the intersection of $\mathrm{SL}_p$ and $Z(\mathrm{GL}_p)$ in $\mathrm{GL}_p$ give you things like $\mu_p$ which is a non-reduced point.
2) This is really quite sophisticated. Since you focused in on the functor of points perspective I'll roughly discuss that. Namely, the naive guess is that the functor of points of $G/N$ should be $(G/N)(R)=G(R)/N(R)$. Unfortunately, this is not the case. 
For example, let's consider the inclusion $\mu_n\hookrightarrow \mathbb{G}_m$ over $\mathrm{Spec}(\mathbb{Q})$ and consider the functor $X:\mathsf{Alg}_\mathbb{Q}\to\mathsf{Ab}$ that takes $R$ to $X(R):=\mathbb{G}_m(R)/\mu_n(R)=R^\times/R^\times[n]$. If this were representable then the map $X(\mathbb{Q})\to X(\overline{\mathbb{Q}})$ would be injective with image precisely $X(\overline{\mathbb{Q}})^{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$. Let's think of the example when $n=2$. Note then that the coset $\sqrt{2}(\overline{\mathbb{Q}}^\times[2])$ in $X(\overline{\mathbb{Q}})$ is not in the image of $X(\mathbb{Q})$ but it is Galois invariant since the Galois conjugates of $\sqrt{2}$ are $\pm \sqrt{2}$ which define the same coset.
In essence, the failure of the equality $X(\mathbb{Q})=X(\overline{\mathbb{Q}})^{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ is an indication that $X$ as a functor $\mathsf{Alg}_k\to\mathsf{Ab}$ is not a 'Galois sheaf' or `sheaf for the etale topology on $\mathrm{Spec}(k)$'. This is bad because all representable things are sheaves for this topology. So what, one does in general is defines $G/N$ not to be the presheaf which takes $R$ to $G(R)/N(R)$ but the sheafification of this presheaf (really we should sheafifify for an even finer topology--the so called fppf topology). This sheafification, as it turns out, is always representable.
As an example, for the case of $\mu_n\hookrightarrow\mathbb{G}_m$ note that we have for every $R$ a natural map $X(R)=R^\times/R^\times[n]\to R^\times=\mathbb{G}_m(R)$ given by $x\mapsto x^n$ which, in fact, is an injection. Now, the failure of $\sqrt{2}(\overline{\mathbb{Q}}^\times[2])$ to be in the image of the map $X(\mathbb{Q})\to X(\overline{\mathbb{Q}})^{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ becomes manifested, as a subset of $\mathbb{G}_m$, as the fact that $2\in\mathbb{G}_m(\overline{\mathbb{Q}})$ is certainly Galois invariant, in the image of $X(\overline{\mathbb{Q}})\to \mathbb{G}_m(\overline{\mathbb{Q}})$ and while it is the image of a point $\mathbb{G}_m(\mathbb{Q})\to \mathbb{G}_m(\overline{\mathbb{Q}})$ (namely $2$!) this point of $\mathbb{G}_m(\mathbb{Q})$ is not in the image of $X(\mathbb{Q})\to \mathbb{G}_m(\mathbb{Q})$ ($2$ is not a square in $\mathbb{Q}^\times$). 
In fact, one sees that the failure of $X(\mathbb{Q})=X(\overline{\mathbb{Q}})^{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$  is somehow minimally fixed by replacing $X$ with $\mathbb{G}_m$. This is, in fact, the idea of the sheafification process and, indeed, the sheafification of $X$ is $\mathbb{G}_m$. 
