The present answer is a try at delineating the use of the verbs that is useful but which might not be very complete; independently of that, some additions not directly concerned with the question have been made with the aim of situating better the forms proposed in the question.
Two verbs are involved, "penser à qqc" and "penser", wherefrom "à quoi" and "ce que" respectively. Here are the TLFi general definitions of the two.
penser à : Appliquer son esprit à un objet abstrait ou concret, présent ou non. Synon. réfléchir, songer (à)
penser : Former des idées; concevoir par l'esprit, par l'intelligence
A. Je veux savoir à quoi tu penses.
1. Two persons are discussing a situation and one of them hinted that some idea about the situation had come to his/her mind; the other person wants to know what it is.
The other person can then say very aptly "À quoi tu penses?" or "À quoi penses-tu?" in a more elegant manner. This a normal way of asking. However, although "Je veux savoir à quoi tu penses." is, according to its construction, a declarative sentence, it has interrogative force, that is, you can use it as it is, without the intonation normally used for questions (and it wouldn't do at all to use that intonation); this is often sufficient to elicit an answer from somebody, specially if the tone ot the elocution is insistant. This does not however confer to the person asking the question a neutral attitude but it shows a certain yearning for an answer. In English we would say "What are you thinking of?".
2. Person A is lost in thought and person B notices the fact, so that out of curiosity for what might be the cause of the unusual state of mind, person B decides to ask person A what he/she is thinking about. In this context also, you can use "À quoi tu penses?" ("À quoi pensez-vous?"). The declarative sentence discussed in the first context is usable too; its use carries the same implications. The English translation is now "What are you thinking about".
B. Je veux savoir ce que tu en penses.
1. Person A is curious about the opinion of person B concerning a given state of affairs; after having related the matter to person B, person A will say in English "What do you think of it?" or "What do you think about it?" or according to the matter, for instance, "What are your thoughts on the subject?". In this context the most apt possibility in French is "Qu'est-ce que tu en penses?" (Qu'en penses-tu?, Qu'en pensez-vous?). The form without "en" will be used in that context but not often and not quite justly.
Notice that what corresponds to "à quoi" is a progressive tense in English and what corresponds to "ce que" is the "state present", but that stems from the verbs.
C. Je veux savoir ce que tu penses.
There is only one context for this construction; it is the general context of thinking where thought may bear upon anything or, on the contrary, upon a restricted subject such as politics, literature, entertainement, the usefulness of a pocketknife and so on. In this context either "Qu'est-ce que tu penses?" and "Je veux savoir ce que tu penses." can't be used in any other way than as elliptic sentences in which the subject is not mentioned, bar exception. That means that in most cases (plain context) the name of a given matter has to be part of the sentence.
1. Plain context _ Qu'est-ce que tu penses du monde de la politique? (de la pâtisserie orientale, des études supérieures, d'avoir des enfants…)
2. Person A is communicating his/her views about a certain subject or situation to person B and at the end of telling person B what his/her thougts are, person A wants to know what are the thoughts of person B on the same subject; in order to do that person A has to say "Que penses-tu?" (no need for the noun of the matter to figure in the sentence); this is the sentence to use so as to refer unmistakingly to person B's thoughts on the subject and to avoid the confusion "What do you think about what I just said?", which would more properly be put as "Qu'est-ce que tu en penses?" and suscitate in person B's mind at the same time as an exposition of his/her views on the subject a critical evaluation of person B's opinions or, alternatively, nothing else but that latter. It must be said that "Qu'est-ce que tu en penses?" could be used for the same purpose but it does not constitute a truly clear-cut question.
In a marginal context now taken up, only the form "Qu'est-ce que tu penses?" can be used and it can still be replaced by "Que crois-tu?"; it is also equivalent to "Qu'est-ce que tu crois?". Its translation in English is the familiar "What do you think?". It is used when trying to make someone become aware of the évidence of something and the need to look no further than plain common sense in order to explain certain facts. Here is an example.
- Qu'est-ce que tu crois? C'est pour se protéger des bêtes sauvages ou des hommes qu'ils construisaient leurs habitats sur pilotis dans les eaux des lacs.