Tensor is sure an important concept in commutative algebra, but the definition is kind of abstract, so is there any way to understand it which is easier? Thanks advance!
The definition I see is the one defined by modules.
Proposition 2.12. Let $M, N$ be $A$-modules. Then there exists a pair $(T,g)$ consisting of an $A$-module $T$ and an $A$-bilinear mapping $g \colon M \times N \to T$, with the following property:
Given any $A$-module $P$ and any $A$-bilinear mapping $f \colon M \times N \to P$, there exists a unique $A$-linear mapping $f' \colon T \to P$ such that $f = f' \circ g$ (in other words, every bilinear function on $M \times N$ factors through $T$).
Moreover, if $(T,g)$ and $(T',g')$ are two pairs with this property, then there exists a unique isomorphism $j \colon T \to T'$ such that $j \circ g = g'$.