There are a lot of different philosophical questions and I'm interested in knowing what kind of questions are asked in or what kind of questions does transcendental philosophy try to answer. I've tried looking but I can't seem to find the answer anywhere.
"Trancendental" is a term used to characterize Kant's epistemology. Kant gives his definition in Critique of Pure Reason (CPR):
I call all knowledge transcendental which deals not so much with objects as with our manner of knowing objects insofar as this manner is to be possible a priori. (B25)
The Kantian term trascendental means to refer to the conditions of the possibility of a science or of a priori knowledge. The term must not be confused with the term transcendent, which means to surmount our empirical experience.
TL;DR: The question that asks what the conditions of the possibility of our reference to objects are. There is only one question in transcendental philosophy, but for aspects or classes of objects it may have to be answered differently.
I would like to make another answer as a sort of appendix to what @JoWehler wrote, based on his letter to Herz in 1772 in addition to CPR, because it is easy to think that one understands what Kant means and what the problem transcendental philosophy tries to answer is, but another thing to actually understand the whole depth of it. As this question is so important and so very underestimated, it will become quite a long answer.
I will translate the letters by myself.
First approach towards the question
The question as asked already occured nine years before CPR and seems rather innocent (Ak. 10:130.6-8):
Ich frug mich nemlich selbst: auf welchem Grunde beruhet die Beziehung desienigen, was man in uns Vorstellung nennt, auf den Gegenstand?"
I asked myself: What is the reason for the relation of what we inside us call representation [Vorstellung] to its object?
Another way to put it, as we will see, is this: How can it be that we think to have representations that actually represent objects? But this is not about the objects of sensual experience, as he continues (lines 8-12):
Enthält die Vorstellung nur die Art, wie das subiect von dem Gegenstande afficirt wird, so ists leicht einzusehen, wie er diesem als eine Wirkung seiner Ursache gemäß sey und wie diese Bestimmung unsres Gemüths etwas vorstellen d. i. einen Gegenstand haben könne.
If the representation only contains the manner in which the subject is effected by the object, it is easy to understand how it, as an effect, corresponds to its cause and how this determination of our mind can represent something, i.e. have an object.
This means he actually takes this to be trivial as there is a causal relation between objects and our sensual representation of them. (Spoiler: Only in the sense of the manifold of intuition).
The other unproblematic case would be if we had an intellect that is productive, an intellectus archetypus, like described in the lines 18-21:
wenn das, was in uns Vorstellung heißt, in Ansehung des obiects activ wäre, d. i. wenn dadurch selbst der Gegenstand hervorgebracht würde, wie man sich die Göttliche Erkentnisse als die Urbilder der Sachen vorstellet, so würde auch die Conformitaet derselben mit den obiecten verstanden werden können.
If that what in us is called representation would be active in respect to the object, i.e. if the object would be produced by it, like divine cognition is thought to be the archetypes of things, the conformity of representations with the objects could be understood, too.
But as finite beings, we have a problem
Now, what Kant is thinking about is a third class of representations as a problem, like also expressed in this letter already (Ak. 10:131.1-10), but it is better put and adressed in CPR, A 92-93/B 124-126 (Cambridge Edition p. 224). I will quote almost the whole paragraph and make comments to make clear where and why a problem occurs:
There are only two possible cases in which synthetic representation and its objects can come together, necessarily relate to each other, and, as it were, meet each other: Either if the object alone makes the representation possible, or if the representation alone makes the object possible.
This corresponds exactly the two cases formulated nine years earlier.
If it is the first, then this relation is only empirical, and the representation is never possible a priori. And this is the case with appearance in respect of that in it which belongs to sensation [i.e. the beforementioned manifold!].
If we talk about sensual representation, it clearly isn't a priori and thus to some extend arbitrary.
But if it is the second, then since representation in itself (for we are not here talking about its causality by means of the will) does not produce its objects as far as its existance is concerned, the representation is still determinant of the object a priori if it is possible through it alone to cognize something as an object.
First, we are here not talking about the divine intellect, but an intellect as ours, a finite one (as becomes clear within further quotes). Therefore the representation does "not produce its objects as far as its existance".
He here restates the problem as follows: Before even being able to produce something by (or after, through the will) thinking it, it has a priori to be thought as object. The problematic relation would therefore be a relation a priori between a representation and an object by virtue of being an object at all:
But there are two conditions under which alone the cognition of an object is possible: first, intuition, through which it is given, but only as appearance; second, concept, through which an object is thought that corresponds to this inuition.
We cannot go depper into why he is able to say this here, just take it for granted for now. It follows from our finite understanding.
It is clear from what has been said above, however, that the first condition, namely that under which alone objects can be intuited, in fact does lie in the mind a priori as the ground of the form of objects. All appearances therefore necessarily agree with this formal condition of sensibility, because only through it can they appear, i.e. be empirically intiuted and given
He here refers to his Transcendental Aesthetics, where he showed space and time as forms of intuition. Now, as objects are given through intuition only, they underlie the same forms. In this sense space and time are the conditions of the possibility of our reference to objects insofar they are given by intuition.
The question now is wether a priori concepts do not also precede, as conditions under which alone something can be, if not untuited, nevertheless thought as object in general, for then all empirical cognition of objects is necessarily in accord with such concepts, since without their presupposition nothing is possible as object of experience.
The question is therefore here for "conditions under which alone something can be [...] thought as object in general":
For they [the concepts *a priori, i.e. the categories of pure understanding] then are related necessarily and a priori to objects of experience, since only by means of them can any object of experience be thought at all.
Where does it get us?
Taken this way,
the transcendental aesthetics are about the conditions of the possibility of our reference to objects insofar they have corresponding intuitions (the forms of intuition, space and time, are the answer),
the transcendental analytic about the conditions of the possibility of our reference to objects insofar they are represented as objects at all (the categories of pure understanding),
and the transcendental dialectics (not mentioned here) about the conditions of the possibility of our reference to objects that do not have a corresponding intuition and are nevertheless necessary (the ideas of pure reason, whose relation to their objects is inexplicable).