Here's how I explain it to my clients who are plaintiffs: When you're the one coming to court asking for relief, it just won't do for you to refuse to cooperate in the pretrial discovery process. A judge who thinks you're cooperating will give you a fair hearing, but if he or she concludes that you're playing games or not cooperating, you're in serious trouble.
In general, and subject (with exceptions) to a six-hour time limit, a witness must answer all questions put to him or her. It is the rule, not the exception, that testimony in disputed cases is likely to be about sensitive topics. It is extremely common for litigants to be probed very thoroughly by opposing counsel on precisely those areas that they'd least like to discuss.
There are two exceptions that may justify refusing to answer specific questions -- and deciding whether they apply is an extremely difficult task even for the most experienced trial lawyers.
Assuming you're in state court (not federal court), Rule 199.5(f) governs when a witness may refuse to answer. (The rule is actually phrased in terms of when a lawyer may instruct a witness not to answer, but if you're pro se you're acting as your own attorney and deemed, for purposes of this rule, to be instructing yourself not to answer.) It says:
"An attorney may instruct a witness not to answer a question during an oral deposition only if necessary to preserve a privilege, comply with a court order or these rules, protect a witness from an abusive question or one for which any answer would be misleading, or secure a ruling pursuant to paragraph (g) [regarding suspending the deposition entirely]. The attorney instructing the witness not to answer must give a concise, nonargumentative, nonsuggestive explanation of the grounds for the instruction if requested by the party who asked the question."
The best-known example of such a privilege is one's right under the Fifth Amendment of the U.S. Constitution to refuse to give testimony that would incriminate oneself of a crime, but there are a very few others. If not asserted in a timely objection, privileges can be easily waived.
Whether a particular question is or isn't "abusive" or "one for which any answer would be misleading" is a difficult judgment call, but judges make those judgment calls with a strong and constant bias in favor of permitting full and open discovery, not protecting anyone's feelings. Refusing to answer an important question typically leads to a motion to compel, possibly coupled with a motion to have the witness held in contempt. And contempt of court proceedings -- which can result in heavy fines and indefinite jail time -- are indeed the ultimate remedy when litigants won't cooperate.
There may be lots and lots of questions whose answers wouldn't waive any privileges, and which aren't abusive, but which nevertheless are objectionable under either the rules of evidence or some principle of substantive law. By default, most of those are "reserved" -- that is, they can be made at, or just before, the trial. Other sorts of objections, however, must be made during the deposition itself, on the record. Knowing when to object, and what objections may be made, and when they must be made lest they be waived, is another one of those things that make lawyers worth hiring. Most pro se litigants fail utterly, or worse, get themselves into trouble, when they try making objections without knowing what they're doing.
It is unlikely that the defendant's attorney will be obviously rude. He or she would be forfeiting a huge advantage that a party with counsel has over a pro se litigant, and trial judges reviewing deposition transcripts take note, and sometimes take offense, when they see that a lawyer has been abusive. Without counsel to help prepare you, and to help represent you during the deposition, you're a soft target anyway. Rudeness isn't needed, and Rule 199.5(d) prohibits it.
Mr. Dyer's answer is outstanding. I am going to assume that you are not going to take his recommendation that you would benefit from hiring an attorney and that you are at a great disadvantage and offer a few more words of warning. In giving a deposition: Always, Always, Always tell the truth; If you can answer a question honestly by simply saying "yes" or "no" do it and shut up; Do not explain your answers unless asked to do so; Do not volunteer information that was not asked for (you will create 3 more questions every time you do this); remember that you are not there to educate or convince anyone of anything; you are there to honestly answer the questions and get through it as quickly as possible; remember as polite and friendly as the attorney asking the questions seems to be (and he/she will be polite) they are not your friend; listen carefully to the question asked and answer it; don't answer the question you thing they should have asked.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.