Question: "First off he said that in general we have a map d:C(V)→M, where M is the cotangent space of all vector fields and C(V) the smooth functions on the manifold, which sends f↦df. He called df "the one form df" and gave no further explanation as to what it is, which is not clear to me. I especially don't understand how it acts on the space of all vector fields."
Answer: When speaking of tangent and cotangent spaces, these notions are defined in terms of the local ring hence you may assyme your scheme is an affine scheme.
If $k$ is a field and $X:=Spec(A)$ is an affine scheme of finite type over $k$, there is a canonical map
$$D: \mathcal{O}_X \rightarrow \Omega^1_{X/k}$$
of sheaves of abelian groups. Here $\Omega^1_{X/k}$ is the sheafification of the module $\Omega^1_{A/k}:=I/I^2$ of Kahler differentials of $A/k$.
Question: "In the end he said that the same construction could be made by tensoring the ring of continuous with itself and then quotient out by an appropriate ideal. Not having understood what came before, this remained mysterious. There is a lot of confusion in my mind right now and if you could clarify even a little bit of this definitions it would help me greatly."
If $m:A\otimes_k A \rightarrow A$ is the multiplication map it follows $I:=ker(m)$ and there is a canonical map
$$d:A \rightarrow \Omega^1_{A/k}$$
defined by $da:=1\otimes a-a\otimes 1.$ The map $D$ is the sheafification of $d$. When $\mathfrak{m} \subseteq A$ is a $k$-rational point it follows the fiber
satisfies
$$\phi:\Omega^1_{X/k}(\mathfrak{m}) \cong \mathfrak{m}/\mathfrak{m}^2.$$
The map $\phi$ is an isomorphism of $k$-vector spaces. There is a canonical map
$$\psi:\Omega^1_{A/k} \rightarrow \Omega^1_{A/k}\otimes_A \kappa(\mathfrak{m}):= \Omega^1_{X/k}(\mathfrak{m}) $$
defined by $\psi(\eta):=\eta \otimes 1$. This is the map your lecturer speaks about.
You map a global section $s\in A$ to the element $\psi(ds):=ds\otimes 1 \in \mathfrak{m}/\mathfrak{m}^2$.
Example: If $\mathfrak{m}_x:=(x_1-a_1,..,x_n-a_n) \subseteq A:=k[x_i]/I(X)$ it follows any section $s\in A$ may be written as
$$s(x_1,..,x_n)=s(a_1,..,a_n)+\sum_i \frac{\partial s}{\partial_{x_i}}(x)(x_i-a_1) + \text{higher order terms}.$$
Hence the term $s(x_i)-s(a_i)$ is in the ideal $\mathfrak{m}_x$. You get canonically an element
$$\overline{s(x_i)-s(a_i)} \in \mathfrak{m}_x/\mathfrak{m}_x^2.$$
Question: "First off he said that in general we have a map d:C(V)→M, where M is the cotangent space of all vector fields and C(V) the smooth functions on the manifold, which sends f↦df. He called df "the one form df" and gave no further explanation as to what it is, which is not clear to me. I especially don't understand how it acts on the space of all vector fields."
This gives a map
$$d_x:\Gamma(X,\mathcal{O}_X)\cong A \rightarrow \mathfrak{m}_x/\mathfrak{m}_x^2.$$
In general for any $Y/k$ there is for any point $x\in Y(k)$ a map
$$d_x: \Gamma(Y, \mathcal{O}_Y) \rightarrow \Omega^1_{Y/k}\otimes \kappa(x) \cong \mathfrak{m}_x/\mathfrak{m}_x^2$$
defined by
$$d_x(s):=d(s_U)\otimes 1$$
where $x\in U:=Spec(A) \subseteq Y$ is an open affine subscheme containing $x$.
Quesiton: "He then went on to say that the cotangent space of all vector fields could also be defined to be generated by the elements "of the form dr" with r an element of the ring of continuous functions, imposing the relations given by Leibniz rule on them."
Answer: The module of vector fields is dual to the module of Kahler differentials:
$$Der_k(A) \cong Hom_A(\Omega^1_{A/k},A).$$
You may define $\Omega^1_{A/k}:= \oplus_{b\in A} Adb/C$ where $C$ is the $A$-submodule generated by $d(\alpha)$ for $\alpha \in k$, $d(a+b)-da-db$ and $d(ab)-adb-bda, a,b \in A$.
See Matsumura's book, "Commutative ring theory" for an introduction.
