I am a noob at derived algebraic geometry and I am very confused at the meaning of `homotopy fiber' of a morphism between two simplicial commutative rings.
In my understanding, this homotopy fiber is defined in the category of homotopy category of simplicial model category sComm, so does it have anything to do with the infinite category structure? Because I always met with such statements like `homotopy fiber in the infinite category Mod(A)' where A is a simplicial ring.
And also is there any concrete way to understand this fiber, or any relationship between this homotopy fiber with the ordinary fiber in the category of commutative rings?
The homotopy fiber is not defined in terms of structure of the homotopy category of the model category $\mathsf{sComm}$, not in the least because the homotopy category often lacks limits, but also because it would not give the homotopy fiber the desired universal property. An important slogan: homotopy (co)limits are not (co)limits in the homotopy category.
The homotopy fiber of a map $f\colon X\to Y$ in an $\infty$-category $\mathscr{C}$ with a zero object $0$ and finite limits is defined to be the pullback $$ \require{AMScd} \begin{CD} \mathrm{fib}(f) @>>> X\\ @VVV @VV{f}V\\ 0 @>>> Y \end{CD} $$ in $\mathscr{C}$ (so this is an $\infty$-categorical limit). It carries a universal property, which essentially states that for any $W\in\mathscr{C}$, the mapping space $\mathscr{C}(W,\mathrm{fib}(f))$ is weakly equivalent to the homotopy pullback $\mathscr{C}(W,0)\times^\mathrm{htpy}_{\mathscr{C}(W,Y)}\mathscr{C}(W,X)$ of spaces. Since $\mathscr{C}(W,0)\simeq *$ by definition of an ($\infty$-categorical) zero object, the universal property actually states that $\mathscr{C}(W,\mathrm{fib}(f))\simeq \mathrm{fib}(f_*\colon\mathscr{C}(W,X)\to\mathscr{C}(W,Y))$ via a specified weak equivalence, where the fiber on the right hand side is a homotopy fiber of spaces (see the last paragraph for an intuitive description how the homotopy fiber of spaces looks like). The fact that $\mathrm{fib}(f)$ is an $\infty$-categorical limit automatically means its definition is stable under equivalences, in the sense that if $f'\colon X'\to Y'$ is a map that is equivalent to $f$ (i.e. there are equivalences $X\simeq X'$ and $Y\simeq Y'$ making the obvious square commute), then the induced map $\mathrm{fib}(f)\to\mathrm{fib}(f')$ is an equivalence.
You cannot define this purely in terms of the homotopy category, because the universal property features a lot of homotopy coherent data for the commutativity of diagrams (you need to supply explicit homotopies witnessing the diagrams commute), while in the homotopy category, commutativity is not data but a property, and moreover the homotopies are not coherent.
Now, in a pointed model category $\mathcal{M}$ (which has an underlying $\infty$-category $\mathcal{M}_\infty$ with a zero object and all limits), we can model homotopy fibers in $\mathcal{M}_\infty$ as follows: if $f\colon X\to Y$ is a fibration between fibrant objects of $\mathcal{M}$, then for any strict pullback square $$ \require{AMScd} \begin{CD} f^{-1}(0) @>>> X\\ @VVV @VV{f}V\\ 0 @>>> Y \end{CD} $$ in $\mathcal{M}$, it holds that $f^{-1}(0)\in\mathcal{M}$ represents the $\infty$-categorical homotopy fiber $\mathrm{fib}(f)$ when passing to the underlying $\infty$-category $\mathcal{M}_\infty$. More generally, given any map $f\colon X\to Y$ between not necessarily fibrant objects in $\mathcal{M}$, we can use fibrant replacements to transform the cospan $X\xrightarrow{f} Y\leftarrow 0$ into a diagram $X'\xrightarrow{f'} Y'\rightarrow 0$ with $f'$ a fibration between fibrant objects, and then the strict pullback $f'^{-1}(0)$ actually also models the homotopy fiber of $f$ in $\mathcal{M}_\infty$.
Why do we need to perform this fibrant replacement, and cannot just take the strict pullback $f^{-1}(0)$ in $\mathcal{M}$ in this latter case? One way to see this is that this would not be stable under weak equivalences in $\mathcal{M}$: if we are some map $f'\colon X'\to Y'$ (not necessarily the same one as above) and weak equivalences $X\to X'$ and $Y\to Y'$ that commute with $f$ and $f'$, then there is an induced map $f^{-1}(0)\to f'^{-1}(0)$, but this induced map need not be a weak equivalence itself. This is one of the original reasons to introduce homotopy (co)limits in model categories, before we had a working theory of $\infty$-categories: in order to make limits and colimits respect weak equivalences in this sense.
You can read up on homotopy (co)limits in model categories in Hirschhorn's book about model categories. You would then also want to read about the Reedy model structures on certain diagram categories.
As for concrete ways to understand the fiber, I would say that you either use its $\infty$-categorical universal property as a way to understand and think about it (it fully characterizes the homotopy fiber up to equivalence), or you think about the homotopy fiber in the model category structure on $\mathsf{sComm}$. This is not an exclusive or, and in fact it probably helps to be fluent with using both. The homotopy fiber in the model category has the benefit that in good cases (when you are taking the homotopy fiber of a fibration between fibrant objects), the strict fiber in the $1$-category $\mathrm{sComm}$ actually models the $\infty$-categorical homotopy fiber as well. In general, as said above, you would need to perform several fibrant replacements (or a single one in e.g. a Reedy model structure on cospan diagrams in your model category) in order to have a strict fiber model the homotopy fiber. The $\infty$-categorical approach has the benefit that it actually characterizes the homotopy fiber with a universal property, which is not possible to fully state in the model categorical world.
As for an intuitive description of the homotopy fiber of spaces (using which you can guess pictures for homotopy fibers in other $\infty$-categories as well): given a map $f\colon X\to Y$ of pointed spaces, the homotopy fiber is a space $F$ consisting of pairs $(x,p)$, where $x$ is a point in $X$ and $p$ is a path from $f(x)$ to the base point of $Y$. A map $(x,p)\to (x',p')$ is the data of a path $q\colon x\to x'$ in $X$, and a homotopy $H$ witnessing that $p'q\simeq p$. Likewise, you can find what higher homotopies are. As you see, we are sort of replacing all equalities you would find in strict $1$-categorical limits with paths, and commutative diagrams with homotopies, and keep on doing this for all higher data as well. This is morally how all homotopy limits of spaces are made.