Asaf's answer is the "right answer", but I'll give a quick (well, as quick as I can) high-level overview of the basic Fraenkel model, which is not a model of ZF where choice fails, but rather a model of ZFA (ZF with atoms/urelements) where choice fails.
The key intuition is that the axiom of choice 'breaks symmetry' when it makes arbitrary choices amongst some symmetric options, whereas all the other axioms preserve symmetry. The reason atoms are useful is that, due to foundation, there are no nontrivial automorphisms of a transitive model of ZF. However, when we have atoms, we can permute the atoms and these permutations induce automorphisms.
We can envision a set as a graph with edges given by the membership relation $\in.$ Then we can unpack the elements of the set as well, and their elements, and so on. This gives a picture of the transitive closure of the set. Well-foundedness implies that the graph has finite depth (although it may be incomprehensibly wide), and each terminal node is an empty set. Atoms are objects that have no elements, but are not the empty set. When we add atoms to the mix, the terminal nodes may be either the empty set or an atom. Formally, we have a picture like the cumulative hierarchy, only instead of starting with $V_0=\emptyset,$ we start with the set of atoms $A,$ and work our way up through higher ranks in a similar fashion.
If we take any permutation $\pi:A\to A$ of the set of atoms, we can extend that to an automorphism of the model by letting $\pi \emptyset = \emptyset,$ $\pi a = \pi(a)$ for any atom, and, recursively, $\pi x = \{\pi y: y\in x\}.$ (In other words, just permute those terminal nodes. Notice particularly that any "pure" set, i.e. one whose transitive closure contains no atoms, is fixed by $\pi$.) Thus we have a group action on our universe that it makes sense to talk about things being symmetrical with respect to.
Consider a universe with a countably infinite set $A$ of atoms, and the group $G$ of permutations of $A.$ For any set $x$, we can define the subgroup we define the subgroup fixing $x$ as $S(x) = \{\pi \in G: \pi x = x\}.$ We will say $x$ is symmetric if $S(x)$ is "large", in a sense we will define precisely. We say a set is hereditarily symmetric if it is symmetric, all of its elements are symmetric, all of its elements' elements are symmetric, and so on. It turns out that the sub-universe consisting of all the hereditarily symmetric sets is a model of ZFA in which choice fails badly.
We define a subgroup $H$ of $G$ to be large if there is a finite subset $E\subseteq A$ such that if $\pi(a) = a$ for all $a\in E,$ then $\pi\in H.$ This $E$ is called a "support" for $H.$ In other words, this subgroup contains all the great many permutations that fix some "small" (i.e. finite) set of atoms. Recall $x$ is symmetric if $S(x)$ is large, so if $x$ is symmetric, we can guarantee $\pi x=x$ by having $\pi$ fix some finite set of atoms.
We will sketch the main results
Theorem. The class of hereditarily symmetric sets is a model of ZFA.
Proof. We will just do pairing. Let $x$ and $y$ be hereditarily symmetric, with supports $E_1$ and $E_2.$ Let $\pi$ fix $E_1\cup E_2,$ which is a finite set. Let $z=\{x,y\}.$ Then $\pi z = \{\pi x,\pi y\} = \{x,y\},$ since $\pi$ fixes the support of both $x$ and $y$. Thus $z$ is symmetric, and both its elements are hereditarily symmetric, so $z$ is hereditarily symmetric. Thus the class of hereditarily symmetric sets is closed under pairing.
All the other closure properties can be proven similarly. The hardest are the ones where there is an arbitrary formula (replacement and separation), but it will be helpful to first prove the fact that $\phi(x,y,\ldots)\leftrightarrow \phi(\pi x, \pi y, \ldots)$
So, the intuition is that doing the usual set theoretical combinatorics with symmetric building blocks only yields symmetric results. (In fact, Jech uses the Godel operation approach, which makes being a model of ZF or ZFA largely a question of closure under set theoretical operations, avoiding pesky formulas.)
Finally, we have
Theorem. The axiom of choice fails over the hereditarily symmetric sets. In particular, the set $A$ of atoms is an infinite set which is Dedekind-finite.
Proof. First, it's not hard to see that $A$ is hereditarily symmetric. Since $A$ is infinite, for any $n,$ there is an injection $f:n\to A.$ We can see this $f$ is hereditarily symmetric as well: $$f= \{(0,a_0), (1,a_1),\ldots, (n-1,a_{n-1})\}$$ so $$\pi f =\{(\pi 0, \pi a_0),\ldots\} = \{(0,(\pi (a_0)), \ldots, (n-1,\pi(a_{n-1}))\},$$ where we recall pure sets, such as the natural numbers, are fixed by any $\pi.$ If we choose $E=ran(f),$ we see that any $\pi$ fixing $E$ fixes $f.$ Thus $f$ is hereditarily symmetric. So for any $n$, there is a hereditarily symmetric injection $f:n\to A,$ thus $A$ is infinite in the class of hereditarily symmetric sets.
On the other hand let $f$ be an injection $f:\omega \to A.$ We can see that $f$ is not symmetric, since for any finite set $E$ of atoms, we can always find a permutation that fixes $E,$ but moves some atom in the range of $f.$ Thus, in the class of hereditarily symmetric sets, there is no injection $\omega\to A,$ i.e. $A$ is Dedekind finite. This implies that $A$ cannot be well-ordered, since otherwise the initial segment length $\omega$ of the well-ordering would be an injection $\omega\to A.$
There is a lot more to study about this model. And this whole process can be generalized by picking different groups of permutations and different "notions of largeness" in order to create other models where choice fails in different ways. (The "notions of largeness" are called normal filters. They are often generated by collections of "small" sets of atoms, generalizing the collection of finite sets used here. These collections of small sets are called normal ideals.)
As I mentioned before, this uses the atoms in an essential way (alternatively you can work in models where foundation fails and use Quine atoms instead of atoms). In ZF, the models you construct are similar, but the big difficulty is where to find the symmetry transformations given that there are no automorphisms of models. The answer involves forcing, so is more technically complicated (thus >30 years between Fraenkel and Cohen), but you can read about it in Jech's book. In fact, the similarity can be made precise in the form of embedding theorems, also covered in the book.
