There are various categories associated with topology. The most fundamental is Top The category of topological spaces.
It has:
In the most general definition the points map like sets and there is an additional requirement that the preimage of an open set is an open set. That is the open sets map in the reverse direction. 
This is a category because the composition of two continuous maps is also a continuous map. The composition is f•g for points and
note: change of order for open sets. 
Also the Id is the Id for points and the Id for opensets. 
The isomorphisms in this category correspond to homeomorphisms in topology.
Top preserves limits and colimits. Simplicial sets are colimits of Top.
Top is a category but it does not have exponential objects for all spaces and so it does not have some nice properties: it is not cartesian closed or a topos. If these properties are required then there may be variations of Top for the spaces required that have these properties. For example, nice topological spaces such as compactly generated topological spaces. Or see homotopy category at ncatlib.
See page here about categorical limits.
There is a category of the subset structure of open sets. There are also categories of fibres and sheaves.
Top  Topological Spaces
Consider a category C as follows:
 Objects are spaces
 Morphisms are structure preserving morphisms between these spaces
For a noncategory theory view of this structure see page here.
Subset Structure of Open Sets as Category
Subsets give interesting structure to open sets. This allows us to define a category of topological spaces. We can also further elaborate this subset structure to get fibre bundles and sheaves.
In this diagram the open set U is a subset of V. UV We can represent this as an arrow from V to U V>U 

This has the properties we expect from a category, for instance, the identity map: U>U (identity map) 

and composition (U>V)*(V >W) = U>W (composition) 
For a noncategory theory view of this structure see page here.
Bn  Category of Bundles
A fibre bundle is a function f:(A>I) This is described on the following pages: 
We can make this a category where:
This triangle must commute. So the elements (germs) of a stalk in 'A' must map to the same stalk in 'B'. This is a comma category as discussed on this page. 
The Bn Category can lead on to the concept of a Topos.
For more information about the Bn category see this page.
Simplical Sets
We have looked at these, mostly from a topology point of view, on the pages here:
The subject can also be approached from a purely combinatorial point of view.
Here we investigate how these structures can be viewed in a category theory way.
Δ is a category with:
The morphisms are inclusions 

Δ^{op} is a category with the same objects but morphisms are the face maps. 
Category of Presheaves
Objects: C^{op} > setcontraveriant functors X: C > set (written X: C^{op} > set to indicate contraveriance) 

Morphisms:are natural transformations N: X > Y 
For a noncategory theory view of this structure see page here.