Top is the category of topological spaces and continuous maps simply by definition; topology typically deals with continuous maps, making this category the most important one, and thus by convention it's the one meant when saying "the category of topological spaces".
(aside: other conventions on what Top or "the category of topological spaces" stands for are far more likely to disagree on what the objects are, rather than the morphisms. e.g. to make the objects be merely the compactly generated Hausdorff spaces)
You can, of course, make all sorts of other categories. The category of topological spaces and open maps is a perfectly reasonable category to make; it's just less useful.
It takes a bit to get used to, but category theory rejects the mindset that mathematics is about objects, with the mappings between objects being a derived notion. Instead, you need to consider objects and mappings as equals -- or even to consider the objects superfluous.
On that last point, my favorite example of a category whose emphasis is on the morphisms is matrix algebra. The set of all matrices, with composition defined by multiplication, form a category. (with addition, you get an Abelian category) The objects of this category really play no role beyond bookkeeping to say which matrix products are defined.
(This category is, of course, equivalent to the category of fintie-dimensional vector spaces and linear maps)
You have the following inclusions:
$$\{ \textrm{inner product vector spaces} \} \subsetneq \{ \textrm{normed vector spaces} \} \subsetneq \{ \textrm{metric spaces} \} \subsetneq \{ \textrm{topological spaces} \}.$$
Going from the left to the right in the above chain of inclusions, each "category of spaces" carries less structure. In inner product spaces, you can use the inner product to talk about both the length and the angle of vectors (because the inner product induces a norm). In a normed vector space, you can only talk about the length of vectors and use it to define a special metric on your space which will measure the distance between two vectors. In a metric space, the elements of the space don't even have to be vectors (and even if they are, the metric itself doesn't have to come from a norm) but you can still talk about the distance between two points in the space, open balls, etc. In a topological space, you can't talk about the distance between two points but you can talk about open neighborhoods.
Because of this inclusion, everything that works for general topological spaces will work in particular for all other spaces, but there are some things you can do in (say) normed vector spaces which don't make sense in a general topological space. For example, if you have a function $f \colon V \rightarrow \mathbb{R}$ on a normed vector space, you can define the directional derivative of $f$ at $p \in V$ in the direction $v \in V$ by the limit
$$ \lim_{t \to 0} \frac{f(p + tv) - f(p)}{t}. $$
In the definition, you are using the fact that you can add the vector $tv$ to the point $p$. If you try to mimick this definition in a topological space, then since the set itself doesn't have the structure of a vector space, you can't add two elements so this definition doesn't make sense. That's why during your studies you sometimes restrict your attention to a smaller category of spaces which has more structure so you can do more things in it.
You can discuss the notions of continuity, compactness only in the category (context) of topological spaces (but for reasons of simplicity it is often done in the beginning of one's studies in the category of metric spaces). However, once you want to discuss differentiability, then (in first approximation, before moving to manifolds) you need to restrict your category and work with normed vector spaces. If you also want to discuss the angle that two curves make, you will need to further restrict your category and work with inner product vector spaces in which the notion of angle makes sense, etc.
Best Answer
Given two structures of $A,B$ of some kind, you first have to guess what a structure-preserving 'map', i.e. a (homo-) morphism $A \to B$ (from $A$ to $B$) is. For the sake of simplicity, we are going to assume, that a morphism is a function with some special properties (this need not be the case, but it very often is). If you have this, then $A$ is commonly called isomorphic to $B$, if there is an isomorphism $A\to B$.
An isomorphism $f : A\to B$ is a morphism $A\to B$ with an inverse, that is a morphism $g : B\to A$ with $g\circ f = \operatorname{id}_A$ and $f\circ g = \operatorname{id}_B$. Of course, in our case we then have $g = f^{-1}$. So, being an isomorphism implies being bijective. The converse is not generally true, because even if $f$ is a bijective morphism, $f^{-1}$ is not necessarily a morphism too (this is indeed the case for topological spaces and their morphisms), but it is true for example for linear maps.
How do you decide what a morphism should be, i.e. what kind of special properties a map between structures should have? Well, it depends. Sometimes its obvious (vector spaces), sometimes it is not (metric spaces). But in any case, the morphisms should preserve "all" the structure. So vector space morphisms (linear maps) should preserve addition and scalar multiplication and a metric space morphism should preserve the metric.
Unfortunately or maybe interestingly, there are many things you could consider "preserving the metric". One kind of possible morphisms are simply called "distance-preserving", but there are other possibilities (short maps, lipschitz-continuous maps, etc.). Because in this case, we have many notions of morphisms, someone decided to use the term "isometric" in place of "isomorphic".
The general theory dealing with "morphisms between structures" is called "category theory".