Help me understand the tensor product

Help me understand the tensor product

I have several books and other literature that define the tensor product,
but I understand none of them. Since this really concerns one topic,
namely understanding the construction of and arithmetic of (with?) tensor
products, I'll pose a few questions here instead of asking them
separately. I don't feel that the specific case of tensoring vector spaces
helps with understanding the general construction, so my questions will
deal with tensoring a right $R$-module with a left $R$-module over the
ring $R$, which we do not assume to be commutative.
1) Now then, the first step is to form the free abelian group $F(M \times
N)$. I've never stumbled upon free abelian groups prior to reading about
the tensor product, but it seems simple enough. Correct me if I'm wrong,
but $F(M \times N) = \{ \sum a_{ij}(m,n) \mid a_{ij} \in \mathbb{Z} \}$,
the sums ranging over all $(m,n) \in M \times N$. But some authors say
something else, which I trust is the same thing but nevertheless confuses
me. Keith Conrad (assuming $R$ is commutative) says $$R=\bigoplus_{(m,n)
\in M \times N} R \delta_{(m,n)}$$ without any mention of what
$\delta_{(m,n)}$ is, but it's probably similar to what I have in a
compendium (which also assumes the ring is commutative), "notation tagging
the component that corresponds to the element $(m,n) \in M \times N$".
What does that even mean, and why is that particular notation only used
when $R$ is assumed commutative?
2) Next define the subgroup $S \subset F(M \times N)$ generated by "all
elements of the following three types" $(a,b+b') - (a,b) - (a,b')$ etc.
Well, what does $-$ mean? Is an equivalence relation meant? If so, when
are they considered equivalent? I've only seen this along the lines of
"define $x,y$ to be equivalent if property $P(x,y)$ holds".
3) $M \otimes N$ is an abelian group. Fine, so how are elements "added"?
No mention of this is made in any of my literature, except that $a \otimes
(b+b') = a \otimes b + a \otimes b'$, but how about $a \otimes b + c
\otimes d$?
4) So-called elementary or pure tensors. Conrad writes "Tensors in $M
\otimes_R N$ that have the form $m \otimes n$ are called elementary
tensors". The elementary tensors span the tensor product (right?), but
what elements of $F(M \times N)$ wind up as elementary tensors? While my
compendium (which, as Conrad, assumes the ring to be commutative) also
makes note of elementary tensors, Rotman ("Introduction to Homological
Algebra"), not assuming commutativity, makes no mention of them, but says:
"Since $A \otimes_R B$ is generated by the elements of the form $a \otimes
b$, every $u \in A \otimes_R B$ has the form $u = \sum_i a_i \otimes b_i$.
This expression is not unique..." The other names for "elementary tensor"
which Conrad lists are not in the index to Rotman's book, so are
elementary tensors only relevant to tensoring over a commutative ring?
Thanks in advance for any help. I've had some homework regarding tensors
and I'm unable to take even a first step in answering the questions, so I
have to learn tensor products somehow.
Conrad: http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf