Challah, kugel, cholent, chicken soup, fish and meat/chicken, etc.
Answer:
Most of our cuisine (once it's kosher) isn't bound by Jewish law. For that reason, there's a lot of variation in such a minor matter as what is served at meals.
1) These are necessary at Shabbat and festival meals: bread and wine. Slightly sweet, braided challah-bread is customary; as is sweet red wine.
2) These are long-established customs: fish (typically gefilte fish, and especially at the evening meal); and cholent at the morning meal. Cholent is a slow-cooked stew of barley, meat and beans, with other ingredients to taste, but each family adjusts the basic makeup of the cholent as they wish.
3) These are common: soup at the evening meal; especially chicken soup (that's the famous "Jewish chicken soup" with its reputed healing-properties). Egg salad with chopped onions, and/or chopped liver.
4) Other dishes, including dessert, will vary according to family or community habits and taste.
The Jews eat the Passover Seder meal on the night of Passover (Pesach). It makes little difference whether Passover begins on Shabbat or on a weekday. Note that there are festive meals on every Shabbat, but they are not called "seder" and the foods are different.
There is no prohibition against eating kosher meat during Shabbat. The reverse is true; it's expected that meat be eaten during the two main Shabbat meals. On Passover, Jews cannot eat leavened bread (whereas they can at other times of the year). Shabbat does not mirror this. The same kosher rules for the rest of the week apply on Shabbat. Jews cannot eat pork on any day of the week and conversely, Jews can eat tuna or kosher meat on any day of the week.
Jews do not celebrate Christmas and so do not eat any special foods for Christmas.
Religiously observant Jews will never eat non-kosher foods, regardless of which day of the week it is. Less observant Jews will make a point to eat kosher food during Shabbat. Non-observant Jews usually don't pay any attention to whether or not the food they are eating is kosher.
Practicing Jews will only eat kosher foods.
They pray, eat, pray, and eat some more. And sit around and shmooze a lot.
Orthodox Jews are only allowed to eat kosher foods.
Muslims and Jews don't eat pork. Muslims eat shellfish, but Jews don't.
Religiously observant Jews will only eat foods that are kosher.
There is no prohibition against eating in a restaurant during Shabbat. Shabbat and kashrut observant Jews would require that the restaurant be kosher. To get around the issue of not being able to handle money during Shabbat, kosher restaurants will pre-sell meals. Additionally, the restaurant would have to be within walking distance.
Regardless of the day of the week, Jews eat only Kosher foods.
Religiously, Jews are supposed to only eat kosher food. That being said, depending on where we live, we have different food styles.