What does birational equivalence signify geometrically?
I have been reading AG and my question is why birational equivalence came into the picture? My basic intuition is we may study a space, $X$ not so much different from the space, $Y$ and we will still be able to absorb great amount of information of the space, $Y$. Because except at certain points they are almost similar(?). For example the affine line and the hyperbola are birational. But! What information in general two birational objects posses?
I have just started reading the subject. Can anyone please explain it to me or provide me some resource?
One important fact towards understanding birational maps is the following, which appears for example in Hartshorne as Lemma V.5.1.
Note for example that this implies that a rational map from a smooth projective curve is always regular. This can be seen in more concrete ways, though. Below I record an example of how this fact can be used to deduce birational invariants of smooth algebraic varieties.
Let $X$ be a smooth projective variety over $\mathbb{C}$. Denote $\Omega_{X}$ the sheaf of algebraic $1$-forms on $X$ and $\omega_X = \det \Omega_X$ its top exterior power. This line bundle is often called the canonical bundle of $X$. Being a line bundle, there is an induced map $\varphi: X \dashrightarrow \mathbb{P}(\Gamma(X, \omega_X)^{\vee})$.
(Explicitly $\varphi$ sends $x \in X$ to the projective equivalence class of the evaluation $\operatorname{ev}_x: \Gamma(X, \omega_X) \to \mathbb{C}$. Note that this is only well defined when $\operatorname{ev}_x$ is not the zero functional: not all global sections of $\omega_X$ vanish at $x$.)
We call $\varphi$ the first canonical map and the (closure of) the image the first canonical image $X^{[1]} \subset \mathbb{P}(\Gamma(X, \omega_X)^\vee)$. As the name and notation suggest, one can repeat this with $\omega_X^{\otimes n}$ to obtain the $n$'th canonical map.
I claim that all of these are birational invariants. Let $f: X \dashrightarrow Y$ denote a birational map of smooth varieties.
Indeed, if $U \subset X$ is the complement of $\operatorname{Ex}(f)$ then there is a birational map $f|_U: U \to Y$ which induces a map $f^*: \Gamma(Y, \omega_Y^{\otimes n}) \to \Gamma(U, \omega_U^{\otimes n})$. But then $\omega_U^{\otimes n} \cong (\omega_X^{\otimes n})|_U$ and since $U$ is the complement of a closed set of codimension $\geq 2$, $\Gamma(U, \omega_U^{\otimes n}) = \Gamma(X, \omega_X^{\otimes n})$.
Hence there is an induced map $f^*: \Gamma(Y, \omega_Y^{\otimes n}) \to \Gamma(X, \omega_X^{\otimes n})$. Symmetrically one can define $(f^{-1})^*: \Gamma(X, \omega_{X}^{\otimes n}) \to \Gamma(X, \omega_Y^{\otimes n})$ and one can check that this is in fact an inverse to $f^*$.
This then induces a linear isomorphism $\mathbb{P}(\Gamma(X, \omega_X^{\otimes n})^\vee) \cong \mathbb{P}(\Gamma(Y, \omega_Y^{\otimes n})^\vee)$ which identifies $X^{[n]}$ with $Y^{[n]}$, up to a linear automorphism of $\mathbb{P}(\Gamma(Y, \omega_Y^{\otimes n})^\vee)$.
Hence to every birational equivalence class of varieties one can associate a sequence of projective varieties $X^{[n]}$ and in fact, for $n$ large and divisible these varieties stabilize to a variety $X^{\text{stab}}$ which we call the stable canonical image. Since the canonical line bundle is the only natural nontrivial line bundle to consider on any smooth variety, this is a rather strong constraint on any given birational equivalence class.
A good elementary overview of this kind of thing can be found in Koll'ar's paper The Structure of Algebraic Threefolds.