Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product, Semi-direct products, so after all why do we need Tensor Product?

The Definition of Tensor Product at Planet Math is confusing.

Definition:  Let $R$ be a commutative ring, and let $A,
B$ be $R$-modules. There exists an $R$-module $A\otimes B$, called the
tensor product of $A$ and $B$ over $R$, together with a canonical
bilinear homomorphism
$$\otimes: A\times B\rightarrow A\otimes B,$$
distinguished, up to isomorphism, by the following universal
property.
Every bilinear $R$-module homomorphism
$$\phi: A\times B\rightarrow C,$$
lifts to a unique $R$-module homomorphism
$$\tilde{\phi}: A\otimes B\rightarrow C,$$
such that
$$\phi(a,b) = \tilde{\phi}(a\otimes b)$$
for all $a\in A,\; b\in B.$ 

The tensor product $A\otimes B$ can be constructed by taking the free
$R$-module generated by all formal symbols
$$a\otimes b,\quad a\in A,\;b\in B,$$
and quotienting by the obvious bilinear relations:
\begin{align*}
 (a_1+a_2)\otimes b &= a_1\otimes b + a_2\otimes b,\quad &&a_1,a_2\in
 A,\; b\in B \\
 a\otimes(b_1+b_2) &= a\otimes b_1 + a\otimes b_2,\quad &&a\in A,\;b_1,b_2\in
 B \\
 r(a\otimes b) &= (ra)\otimes b= a\otimes (rb)\quad &&a\in A,\;b\in
 B,\; r\in R
\end{align*}

Also what do is the meaning of this statement:(Why do we need this?)


Every bilinear $R$-module homomorphism
$$\phi: A\times B\rightarrow C,$$
lifts to a unique $R$-module homomorphism
$$\tilde{\phi}: A\otimes B\rightarrow C,$$
such that
$$\phi(a,b) = \tilde{\phi}(a\otimes b)$$
for all $a\in A,\; b\in B.$


Me and my friends have planned to study this topic today. So i hope i am not wrong in asking such questions.

Tensor products are useful because of two reasons:


they allow you to study certain non linear maps (bilinear maps) by transforming them first into linear ones, to which you can apply linear algebra;
they allow you to change the ring over which a module is defined.


There are many, many ways in which these two abilities show up in nature.
