As I listen, I often hear stations calling "CQ contest". When I hear stations respond, they seem to do so with a specific set of information. I recognize the call sign and a signal report but there are often other pieces of information that seem to be standard that I don't recognize. When I hear a CQ contest call I am not sure how to respond.
A "CQ Contest" is simply a limited CQ, just like someone can call CQ some-specific-area or CQ any-member-of-a-particular-club. I don't think I've heard this variant on SSB, but I imagine in some contests it may be beneficial to actually specify the contest in question.
During a contest, there is usually a specific set of information to exchange. The exact information to be exchanged varies with the contest. This can be for example a serial number, signal report, Maidenhead locator, state/province/county/etc. (usually according to some sort of list of codes), or almost anything else, or a combination of several of these.
In order to participate, you must know what the specific exchange is for the contest that the other station is calling CQ for. If you don't know what the exchange is and what yours should be, don't respond to a CQ Contest call. The only exception I can think of is if the remote station is calling repeatedly but not getting any answer; in that case, you might respond with something brief like "W9ABCD, ABC1DE what's the exchange?", substituting the correct call signs. If it's simply a serial number (or something similar that you know right away) and signal report, you can jump right in and figure out later where to send the logs. Don't ask such a question on the air if the remote station is working stations; you almost certainly will annoy everyone involved. If the exchange is only a signal report and a serial number, by listening to a few exchanges you should be able to figure that out quickly enough; if it's anything much more involved you'll probably need to look it up somewhere anyway. If the remote station is weak, it may be beneficial to listen to a few exchanges anyway in order to establish the remote station's transmitted exchange before you try to contact it yourself.
A contest QSO is just like any other QSO, only in fast-forward mode. A typical phone contest QSO might go something like this:
W9ABCD transmits: W9ABCD CQ contest
N5EFGH transmits: N5EFGH
W9ABCD transmits: N5EFGH five-nine, one-four-zero
N5EFGH transmits: Five-nine, zero-one-eight
W9ABCD transmits: Roger, contest
...at which point some other station hopefully responds to W9ABCD, and N5EFGH writes down the QSO in the log and moves on to work the next one. Working a contest by scanning the bands and working stations encountered is called search and pounce, by the way. By only giving the absolute bare necessities, a strong-signal contact can be over in a few seconds.
Note that to save time, the signal report for any audible contest signal is usually 59 (or 599 on CW and digital modes). That means that you'll normally only need to write down the exchange received (since you know your own transmitted exchange).
On non-phone modes, the exact traffic will obviously differ slightly from the above example, but the general idea remains the same: getting the particular contest's exchange well, exchanged, as quickly and efficiently as possible.