Probing in a survey is a technique used where the interview is administered by a interviewer (i.e., face-to-face or by phone) and the interviewer prompts the respondent for more information to a question. For example, the question may ask "Why did you buy this product?" and, once the respondent has given an initial answer, the interviewer may probe for more information by asking "Are there any other reasons" a few times.
In qualitative research probing is conducted in a more conversational fashion, with the interviewer asking follow-up questions designed to obtain more and deeper information. See mktresearch.org for an example.