This answer is an elaboration on Dylan's comments.

1. Let us define a homotopy theory to be a pair $(C, W)$, where $C$ is a category and $W$ is some class of morphisms called weak equivalences. (Let's say that $W$ satisfies some reasonable properties, e.g. it's contains isomorphisms and is closed under composition.)

Of course, the archetypal example is given by topological spaces with the class of weak homotopy equivalences, which are continuous maps that induce isomorphisms on homotopy groups. This is the homotopy theory that has been studied classically by algebraic topologists. Similarly we may consider the category of simplicial sets, again with the class of weak homotopy equivalences. Another classical example of a different flavour is given by the category of chain complexes of $R$-modules, for a commutative ring $R$, together with the class of quasi-isomorphisms (morphisms which induce isomorphisms on homology groups). This homotopy theory is also known as homological algebra.

2. Now, we would somehow like to express the idea that the homotopy theories of topological spaces and simplicial sets are equivalent, even though the categories themselves are far from being equivalent. That is, we would like to equip the category of homotopy theories itself with a class of weak equivalences (I'm going to ignore size issues here).

One idea is to consider Gabriel-Zisman localization: given any homotopy theory $(C, W)$, there is a canonical construction $C[W^{-1}]$ which universally inverts all morphisms in $W$. Thus we could say that an equivalence of homotopy theories is an equivalence of the associated Gabriel-Zisman localizations. The homotopy theories of topological spaces and simplicial sets will then be equivalent in this sense, according to a theorem of Milnor. The downside of this definition is that any homotopy theory $(C,W)$ will be indistinguishable from the homotopy theory $(C[W^{-1}], isos)$ (where we take weak equivalences to be isomorphisms). Experience has taught us that is not what we want: the construction $C[W^{-1}]$ is poorly behaved from the homotopical point of view, as it is not even possible for instance to recover homotopy (co)limits from $C[W^{-1}]$, while ideally we would expect that any homotopy theory should have an internal notion of homotopy (co)limits.

Instead, one should use a much more refined version of the construction $C[W^{-1}]$, namely the $(\infty,1)$-categorical localization; it can be modelled for example by the Dwyer-Kan localization of simplicially enriched categories. This gives a good notion of weak equivalence of homotopy theories. We then have a striking theorem of Clark Barwick and Dan Kan that can be paraphrased as follows:

The homotopy theory of homotopy theories is equivalent to the homotopy theory of $(\infty,1)$-categories (as modelled by quasi-categories, complete Segal spaces, simplicially enriched categories, etc.).

We can then view the pair $(C,W)$ as a presentation or model of the associated $(\infty,1)$-category. For example, the homotopy theories of topological spaces and simplicial sets are both models of the same $(\infty,1)$-category.

3. The theorem of Barwick-Kan tells us that, from the perspective of homotopy theory, there is no difference between homotopy theories or say, quasi-categories. However, from the perspective of category theory, these two models are very different.

The question is about how to access categorical information in a given homotopy theory $(C,W)$. That is, in ordinary category theory, we are used to talking about objects, morphisms, functors, limits and colimits, presheaves, the Yoneda lemma, and so on. We have $(\infty,1)$-categorical versions of all these things, but how do we see them inside a given pair $(C, W)$? Of course, we may always look at the respective operations in the underlying category $C$, but these will in general not be compatible with our class of weak equivalences (e.g. the (co)limit of two weakly equivalent diagrams may not be weakly equivalent).

We can view the theory of model categories as a solution to this problem: the idea is to endow the pair $(C,W)$ with a model structure, i.e. the structure of cofibrations and fibrations in an appropriately nice way. Doing this is usually nontrivial, but when possible, it gives us a powerful way to compute things like homotopy (co)limits, namely by computing them in the underlying category $C$ after taking suitable (co)fibrant replacements.

We should keep in mind though that what we really care about are homotopy theories. Despite the effectiveness of model categories, it is a fact that the choice of specific model-categorical presentation adds a factor of arbitrariness to all constructions and proofs, and as a result does not always allow us to express ourselves quite as fluently as we are used to in ordinary category theory. Nowadays it is also common to work with other models of $(\infty,1)$-categories instead of working directly with homotopy theories; for example, the model of quasi-categories, developed by Joyal and Lurie, has an amazingly well-behaved category theory (even if it comes with its own difficulties). At the end of the day, these are all just equally valid approaches to working with homotopy theories.