I'm reading chapter 6 about Teichmüller space in the book: Geometry and Spectra of Compact Riemann Surfaces by Peter Buser. In the following definition:
Definition 6.1.2: Two marked Riemann surfaces (S,$\phi$) and (S',$\phi$') are marking equivalent if there exists an isometry $m: S \to S'$ such that $\phi$' and $m\circ \phi$ are isotopic.
I don't understand why he gave this definition and the meaning behind it. Could you please give me the motivation and an intuitive meaning of it. I really appreciate it.
To answer the first part of the question, it might be interesting to point out that it is a definition of Teichmüller spaces. Teichmüller space is quite interesting on it's own and his currently studied by a variety of mathematicians in many different fields. The reason to study that space might change depending on who you would be asking. One reason is because it is the universal cover of the Moduli space. Indeed, there is an action of the mapping class group on Teichmüller space whose quotient is the Moduli Space, which in the terms you used would be the set of Riemann surfaces up to isometry. So let's try to demystify this definition.
The first to unpack is probably the notion of marking in this definition. To keep it simpler, let's assume we are considering closed Riemann surfaces of finite type. Pick a topological surface $S_g$ of genus $g$. A marking $\phi$ on a Riemann surface $S$ is a diffeomorphism $\phi: S_g \to S.$ For two marked Riemann surfaces to be equivalent, there must be an isometry $m : S \to S'$, but such that $\phi' \simeq m \circ \phi$. In particular, you require your isometry to be isotopic to the change of marking. In particular, if you take a Riemann surface with two different marking, which you obtain from doing a Dehn twist along an essential simple close curves, then they won't be equivalent as marked Riemann surfaces. In a maybe more intuitive level, you are studying complex structure that you can put on a topological surface, but you care about what happens to simple closed curves on it as well. Hopefully, this might help you understand the definition better.
I don't know your mathematical background, but let me give you a different definition for Teichmüller space. You can replace the marked Riemann surface by marked hyperbolic surface, namely a diffeomorphism $\psi : S_g \to X:=\mathbb{H}/\Gamma,$ where $\Gamma$ is a subgroup of isometry acting freely and properly discontinuously on $\mathbb{H}.$ Two marked hyperbolic surfaces $(X_1,\psi_1)$, $(X_2, \psi_2)$ are equivalent if there exist an isometry $m : X_1 \to X_2$ such that $\psi_2 \simeq m \circ \psi_1$ or equivalently, $m \simeq \psi_2 \circ \psi_1^{-1}.$ This definition is equivalent to the one you gave because of the uniformization theorem. Indeed, any Riemann surface of genus $g \geq 2$ $S$ as universal cover conformally equivalent to $\mathbb{D}.$ Moreover, $S = \mathbb{D}/\Gamma'$ where $\Gamma'$ is a group of biholomorphic automorhism of $\mathbb{D}$. Such maps preserves the hyperbolic metric and hence this gives you a hyperbolic structure. So an other way to understand this space is as marked hyperbolic structure. Studying Teichmüller space is then studying all those marked structures on a topological surface. The marking is then telling you that if you twist along a simple closed curve, you should obtain something different from what you started, even though there is an isometry between them. I think there is no better way to visualize that than this explanation from, I think, Bill Thurston (however, I am paraphrasing here):
The importance of the marking should be obvious to anyone who tried to put a pajama to a baby. If when you put the pajama, one of the leg of the pajama is all twisted, then the baby won't be confortable, even though it is the same pajama.