This is the sentence where I found it: Πιστεύω υπάρχει μια εφαρμογή που μπορεί να το εντοπίσει.
Then I typed in "What do you think?" in Google translate and it came up with - Τι νομίζετε;
I checked the meanings of both words in the WR dictionary and Wiktionary, and they both said "to think / believe", so I'm actually just wondering about how these meanings compare for these two words in general.
"Πιστεύω" normally expresses a higher degree of certainty than "νομίζω". However, sometimes they may be used interchangeably, specially in spoken language. "Πιστεύω" roughly corresponds to "believe" and "νομίζω" to "think", but I can't be totally sure about the use of English verbs.
In English, the verb πιστεύω can ( sometimes) also be rendered with 'trust' (as in "I trust that..."), since it is connected with the noun 'pistis(AG)/pisti(MG) = faith. Therefore originally "I have faith" (that...).
In that case, I guess that "I trust that ..." can indeed be rendered with "πιστεύω ότι ...", but the verb "trust" mainly corresponds to "εμπιστεύομαι"(German "vertrauen"), which also comes from "πίστις".
I don't quite agree with Konstantinos. I think both sentences he marks as wrong or unidiomatic are perfectly possible:
Αυτή είναι μια πολύ δύσκολη κατάσταση και χρειάζεται πολύ λεπτομερή ανάλυση. Τι νομίζεις;
Εχθές το βράδυ για μια στιγμή νόμισα ότι δεν με αγαπάει.
Wouldn't you say "νόμισα πως ήρθε το τέλος μου"?
But of course πιστεύω σε is the only possible equivalent for 'I believe in'.
Tι λες; is also a possible equivalent of "What do you think?"