The context: last year, I saw this sign that announced the time and place that a Jewish Seder (probably Yom Kippur) would happen. I was definitely curious and considered going and at least observing because as a Christian, my religion has Jewish roots. I ended up not going, mostly out of laziness (I'm a college student, after all), but also somewhat because I didn't know if it was appropriate for me, a Gentile, to observe or participate. Now Passover is coming up and I'm curious again - what is appropriate for me to do as a Gentile at a Seder?
First, about Yom Kippur: A seder, and Passover, have nothing practical to do with Yom Kippur. Now to your question:
There's nothing wrong with a gentile's attending a seder. Obviously, it would be tactless to bring up Jesus's last supper, or any comparison with Easter. (Moreover, it would go directly against one of the main purposes of the seder, which is the Jewish education of Jews, especially of Jewish children. For the same reason, even though you may see them asking questions, you should save yours for another time, so as to allow them their night.)
That said, I suspect a number of Jews would be rather uncomfortable having a religious Christian, specifically, attending their seder, knowing that he's comparing it in his mind to Jesus's last supper. I doubt that that would apply to other gentiles' attendance.
Also, we're not allowed to invite (or, I think, expect) gentiles to attend a holiday meal. (Mishna B'rura 512:3.) If you show up unannounced, that's fine, but only if you show up after the food is all cooked. (MB :10.) What this means practically for you is that (a) you should show up unannounced. Make sure it's the house of someone you feel comfortable showing up at unannounced, of course. But, also, (b) you should show up after the food is all cooked. I don't know the particulars of this rule — it makes sense to me that you could show up while things are still simmering as long as the active part of the cooking is done, but I'm not sure that that's actually the rule — and you may get turned away at the door (or asked not to eat) for this reason: at a seder, the meal is after quite a bit of other goings-on. (Try not to be offended.) However, my understanding (no source) is that this rule doesn't apply if the seder is Friday night: that we're allowed to invite gentiles to the meal in that case.
(Added in proof, so to speak: Others have pointed out in comments that at least some hold that one should not give matza to a gentile to eat. (Taz, OC 167:18. Thanks, Michoel and Fred!) So you'd not get a piece of the matza at the eating-matza parts of the seder.)
Another thing to bear in mind, though, is that a traditional seder is a good few hours long. I find it interesting and enjoyable, but you may find it boring, in part because much of the discussion (depending on the seder you attend) will assume some prior knowledge, whether of Judaism or of Hebrew, you may not have.
But most of this is just my opinion.
R. Moses Feinstein, one of the leading twentieth century rabbinic authorities, wrote a responsum (Igrot Moshe Y.D. 2:132) from which we can perhaps infer an answer to your question. The case he was dealing with was where a Jewish man had married a Gentile woman who had converted under the auspices of a Reform rabbi. R. Feinstein ruled that the conversion was invalid and that therefore the woman was still absolutely a Gentile.
The question then arose as to what someone should do if his relative who is married to this gentile comes to the Passover seder. R. Feinstein's answer only dealt with the potential issues of teaching Torah to Gentiles and the status of the wine, with not even a hint that there might be an issue with having the gentile present in the first place.
Even if R. Feinstein assumed that the querent had no control over the Gentile's presence, you might still expect him to mention it if it was truly an inherent problem. From his silence on this matter we can perhaps infer that in his view there is no inherent issue with a Gentile attending a Passover seder.
There is absolutely nothing wrong with a non-Jewish person attending a seder, if all of the food has been cooked in advance of Yom Tov. As stated above any prohibition that might exist has nothing to do with the seder per se, it has to do with restrictions about holiday preparations. There are a zillion reasons to mitigate this prohibition. Perhaps a nonJew should not eat the "afikoman" as this recalls the Passover sacrifice, which was only shared by Jews and non-Jews (and even some Jews) were not permitted to eat it.