The term ‘topological category’ is traditional, and comes from the frequent examples in topology. It does not mean an internal category or enriched category in Top! (Fortunately the term topological groupoid is not taken by this tradition; indeed, the only groupoid that is a topological category over $Set$ is trivial. On the other hand, they do seem to use the term ‘topological functor’, which here we avoid.)
A topological category is a concrete category with nice features matching the ability to form weak and strong topologies in Top.
Most generally, the definition relates to a functor $U\colon C \to D$ (such as the forgetful functor from $Top$ to Set), but one can think of this as giving $C$ as a bundle over $D$. Sometimes, when $D$ is in fact Set, the category $C$ satisfying the properties described belows is called a topological construct (Preuss). Usually $C$ and $D$ will be large categories. Let a space be an object of $C$, an algebra be an object of $D$, a map be a morphism in $C$, and a homomorphism be a morphism in $D$. (The reason is that, typically, $C$ will be a category of spaces with some kind of topological structure while $D$ will be, if not $Set$, then some kind of algebraic category.)
Then $C$ is a topological category over $D$ if, given any algebra $X$ and any (possibly large) family of spaces $S_i$ and homomorphisms $f_i\colon X \to U(S_i)$ (that is, a ”$U$-structured” source from $X$), there exists an initial lift, which is to say
Here are some illustrative commutative diagrams (if you can read them):
It follows by a clever argument that $U\colon C \to D$ must be faithful; see Theorem 21.3 of ACC. That is also often included in the definition, in which case the uniqueness of $n$ can be left out. Thus we may think of objects of $C$ as objects of $D$ equipped with extra structure. The idea is then that $T$ is $X$ equipped with the initial structure or weak structure determined by the requirement that the homomorphisms $f_i$ be structure-preserving maps.
The dual concept could be called a cotopological category. However, this is not actually anything new; $U\colon C \to D$ is topological if and only if $U^op\colon C^op \to D^op$ is. This is a categorification of the theorem that any complete semilattice is a complete lattice. Thus, every topological category also has final (not usually called terminal) or strong structures, each determined by a family of homomorphisms $f_i\colon U(S_i) \to X$ (a $U$-structured sink to $X$).
Both of these results (faithfulness and self-duality) depend on the fact that we have allowed the family $\{S_i\}$ to be potentially large. Counterexamples are easy to find. For instance, if $C$ is a large category with all (small) products, then the functor $C \to 1$ to the terminal category satisfies the above lifting property for small families $\{S_i\}$. However, it need not satisfy the dual property (unless $C$ also has all small coproducts) nor need it be faithful.
It also follows that $U$ is a fibration and opfibration, in the weakened bicategorical sense of Street. One also often assumes in the definition $U(T) = X$ and that $g$ is the identity morphism, which in particular makes $U$ into a fibration in the original sense of Grothendieck. This is a bit evil, but it is convenient and satisfied in almost all examples, and any example not satisfying it is equivalent to one which does (via fibrant replacement by an isofibration).
The name ‘topological category’ comes from these examples from point-set topology; these are all topological over Set:
In contrast, the category of locales is not topological over $Set$, apparently not even the category of spatial locales (equivalent to the category of sober spaces), essentially because soberification of a topological space may not preserve the underlying set.
Also, the category Diff of smooth manifolds is not topological but most categories of generalized smooth spaces are.
Outside of topology, the category of measurable spaces is topological over $Set$.
The category of topological groups is topological over Grp, the category of topological vector spaces is topological over $\mathbb{R}$-Vect, etc.
If $C$ is topological over $D$, then so is any full retract of $C$, as long as the functors involved live in $Cat/D$.
In particular, a reflective or coreflective subcategory of $C$ is topological, as long as the reflectors or coreflectors become identity morphisms in $D$.
The forgetful functor $U\colon C \to D$ is not only faithful but also (for different reasons) essentially surjective. Thus it is never full (except in the trivial case where $U$ is an equivalence, of course).
If $D$ is complete or cocomplete, then so is $C$.
If $D$ is total or cototal, then so is $C$; see solid functor.
If $D$ is mono-complete or epi-cocomplete, then so is $C$.
If $D$ is well-powered or co-well-powered, then so is $C$.
If $D$ has a factorization structure for sinks $(E,M)$, then $C$ has one $(E',M')$, where $M'$ is the collection of morphisms in $C$ lying over $M$-morphisms in $D$, and $E'$ the collection of final sinks in $C$ lying over $E$-sinks in $D$. This generalizes the lifting of orthogonal factorization systems along Grothendieck fibrations.
If $D$ is concrete, then so is $C$. More generally, if $D$ has a generator, then $C$ is concrete over $D$.
In particular, if $D$ is Set, then $C$ is a concrete category that is complete, cocomplete, well powered, and well copowered.
If $X$ is any algebra, then there is a discrete space over $X$ induced by the empty family of maps. Similarly, we have an indiscrete space with the final structure induced by no maps. This defines functors $disc, indisc\colon D \to C$ that are respectively left and right adjoints of $U$.
Suppose that $D$ has an initial object $0_D$. Then the discrete space $0_C$ over $0_D$ is initial in $C$. Similarly, the indiscrete space over a terminal object in $D$ is terminal in $C$.
More generally, suppose that $D$ has products or coproducts (indexed by whichever cardinalities you may wish to consider). Then $C$ also has (co)products, lying over the (co)products in $D$, with structures induced by the product projections or coproduct inclusions.
More general limits and colimits are constructed in a similar way. However, it is not typically the case that $U$ creates (co)limits in $C$ because creation of a limit requires that every preimage of the limiting cone is limiting. This fails for $U: \mathrm{Top} \to \mathrm{Set}$ since we can coarsen the topology on the limit vertex to obtain a counterexample.
If a single algebra $X$ has been given the structure of several spaces, then there are a supremum structure and an infimum structure on $X$ induced (as the initial and final structures) by the various incarnations of its identity homomorphism. Exploiting this shows how to construct final structures out of initial ones and conversely.
If $X$ is a regular subalgebra of some $U(S)$, then the inclusion homomorphism makes $X$ into a subspace of $S$, which is also a subobject in $C$. Every regular subobject of $S$ is of this form; note however that there may be nonregular subobjects in $C$ even if all subobjects in $D$ are regular.