I know that the Tensor Product is a well structured mesh of concepts, which relates the notion of Universal Property (UP) (which defines the required algebraic structure) and the existence of such space that "satisfies" the UP (the quotient vector space given by $V \otimes W =: F(V \times W)/ I$).
I would say that I simply do not understand the deep reasons of why we want to use free vector spaces and the space $I \subset F(V \times W)$ more than the intuitive notion of "construct a bilinear operation $\otimes$"; this, however, is a too long consideration to translate into a question.
What I want to know is the relationship between the basic quotient set and the quotient vector space $V \otimes W =: F(V \times W)/ I$. I'm going to explain better:
In set theory then, we have a set $A$. Meanwhile we say that a equivalence relation "$\sim$" is a binary relation which satisfies the properties:
$1) x \sim x \hspace{5mm}\forall x\in A$
$2) x \sim y \hspace{5mm}\forall x,y\in A$
$3) x \sim y\hspace{2mm}and\hspace{2mm}y \sim z \hspace{2mm}then\hspace{2mm}x \sim z \hspace{5mm} \forall x,y,z\in A $
Now, given $\sim$ in a set $A$, we construct another set, $\bar{x}$, called the equivalent classes of an element $x \in A$:
$$ \bar{x} =: \{a \in A \mid a \sim x\} \subset A \tag{1}$$
And finally the quotient set
$$ A/\sim =: \{\bar{x} \mid x \in A\} \tag{2}$$
Now, I know that the elements of $V\otimes W$ are $v\otimes w$ which are the equivalence classes of $F(V\times W)$. So clearly, considering $(1)$, we have:
$$ v\otimes w =: \{a \in F(V\times W) \in \mid a \sim (v,w)\} \subset F(V\times W) \tag{3}$$
Well, I don't know what suppose to enter in $a$ (which symbol). Also, consider $(2)$; it's clear that $A \equiv F(V\times W)$, $\bar{x} \equiv v\otimes w$ and $x \equiv (v,w)$. But $\sim$ is a relation not a set (i.e. $\sim \neq I$), and I don't know what suppose to enter in $\sim$, i.e.:
$$ F(V\times W)/\sim =: \{ v\otimes w \mid (v,w) \in F(V\times W)\} \tag{4}$$
If $R$ is a ring and $I$ is an ideal, then we can define an equivalence relation $\sim$ on $R$ by $r \sim s$ if $r - s \in I$. The equivalence class of $r \in R$ is $[r] = \{r + i \mid i \in I\}$ which is also denoted by $r + I$ and called a coset of $I$. The collection of cosets of $I$ is denoted $R/I$, which is nothing but $R/\sim$. Because $I$ is an ideal, $R/I$ inherits a ring structure from $R$: addition is given by $(r + I) + (s + I) = (r + s) + I$, so $I$ is the additive identity and $-(r + I) = (-r) + I$, while multiplication is given by $(r + I)(s + I) = (rs) + I$. If $R$ has multiplicative identity $1$, then $R/I$ has multiplicative identity $1 + I$.
In your case, $v\otimes w = [(v, w)] = \{(v, w) + (x, y) \mid (x, y) \in I\} = (v, w) + I$.