- The first food on the Kaarah (Seder Plate) is the karpas. Some hold that it's one of the mitzvot of the night to demonstrate slavery and freedom, while others hold that it's only brought to get the children to ask. 
Type of Vegetable
- It's preferable to have celery as Karpas. However, there are differing minhagim and some of them include potato, radish, or parsley. Nonetheless, one should be careful to check for bugs, and if one is unable to check, one can fulfill their obligation with any vegetable, yet one should be careful it has the bracha of HaAdama. 
- Rambam holds that it can be whatever one has 
Size of Karpas
- One should eat a piece of Karpas that's smaller than a Kezayit so as not to get into a dispute about Bracha Achrona.  If a person is very hungry one may continue to eat karpas in a way that there’s no mental break between the karpas and maror, avoiding the dispute about reciting borei nefashot. 
Leaning during Karpas
- There's a dispute whether one shouldn't or should lean during Karpas and it's preferable not to lean, but one if one wants one is permitted. 
- When one makes the bracha on the Karpas, one should have in mind to cover the Maror. 
- It's preferable that one person make the bracha for everyone and everyone can listen. However, if the one making the Bracha doesn't say the words accurately or won't have the proper intent to fulfill everyone else, everyone should make their own bracha. 
- Karpas: The Vegetable at the Beginning of the Seder by Rabbi Josh Flug
- The simple explanation of the Gemara Pesachim 114 is that we dip the Karpas to get the children to ask. However, Rambam holds that one has to eat a Kezayit of the Karpas implying that the Karpas has the status of a mitzvah of the night. See Rav Yosef D. Soloveitchik (Haggadat Si'ach HaGrid no. 5) who explains this more fully. The other aspects to Karpas is sourced in the Minhagim (Haggadah 16) who says that Karpas is eaten before the meal to act like royalty who serve appetizers before a meal. Lastly, the Avudraham (Seder HaHaggadah s.v. Vekatav HaRaavad) writes that Karpas represents the idea of slavery through the letters of Karpas spelled backwards: "Samech" which is equal to 60 (corresponding to the 600,000 who left Egypt) and the Avodat "Parech" (back-breaking labor). This can also be reflected in the practice to dip Karpas in salt water.
- The Gemara Pesachim 114a doesn't identfy what vegetable should be taken as the vegetable dipped before the meal. Manhig (Siman 60) writes the Minhag is to do it with Karpas because of the drasha on the letters of Samach Parech. Arizal (quoted by Od Yosed Chai Tzav 6, Kaf HaChaim 473:14, Chazon Ovadyah (Pesach vol 2 pg 34)) writes that the minhag of using Karpas is established and shouldn't be changed. Rashi (Sukkah 39b) says that Karpas could be translated as cress (similar to parsley) but he heard (from his rabbis) that Karpas is translated as Apie. Sh"t Chatom Sofer 132 in the name of his teacher Rabbi Natan Adler says that the Apie vegetable is celery. Machasit haShekel (473:4) in name of medical books, and Sh"t Bet Shearim 213 concur that Karpas is celery. Halichot Shlomo writes that Rav Shlomo Zalman Auerbach when he was younger ate celery and in his later years had cucumber as Karpas.
- Peirush Hamishnayos (Pesachim 10:3)
- Bet Yosef brings a dispute between the Rosh, Sh"t HaRashba, Tur, Hagahot Maimoniot, Hilchot Chametz U'Matzah 8:4 who hold that less than a Kezayit of Karpas is needed because the Karpas is only meant as a way to get the children to ask. However, Rambam (Hilchot Chametz U'Matzah 8:2) holds that one must have a Kezayit of karpas. [Another Rishon, the Rashbetz (Yafeh Shamuah pg 35) holds like Rambam because the washing of Urchatz only applies to dipping a vegetable of a the size of a Kezayit.] Shulchan Aruch 473:6 rules like the Rosh against Rambam because of the issue brought up by the Sh"t Maharil 25. There's a dispute between Rashbam (Pesachim 114b s.v. peshita), who holds the Bracha on Karpas covers the Maror, and the Tosfot (Pesachim 115a s.v. VeHadar in name of the Ri), who holds that the Maror doesn't need a bracha as it's part of the meal. This is why the Tur (473:6) writes that according to Rashbam one should not recite borei nefashot after karpas even if one did in fact eat a k'zayit, while according to Tosfot you would. Therefore, suggests the Maharil (Responsa 25), in order to satisfy the opinion of the Rashbam one should have less than a Kezayit so that one doesn't have to make a Bracha Achrona before the meal and put oneself in doubt about the Bracha on Maror. Shulchan Aruch 473:6 rules in accordance with Maharil. This is also the opinion of Mishna Brurah 473:53 and Sh"t Chazon Ovadyah 17. See further concerning the significance of Karpas at YUTorah by Rabbi Josh Flug
- Rav Hershel Schachter (“Inyonei Pesach 4”, min 54-5)
- Mishna Brurah 473:56 writes that even if one did eat more than a Kezayit one should not make a Boreh Nefashot because we want the HaAdama to cover the Maror. However the Gra holds that since the Ashkenaz Minhag is to make a bracha on each cup of wine, it must be that we hold that the Haggadah is an interruption and if so, one should make a Boreh Nefashot. Nonetheless, Beiur Halacha says that the Magen Avrham would argue that the reason we make a bracha on each cup is because each is it's own mitzvah and one can make the same claim that Karpas and Maror are separate Mitzvot. Chazon Ovadyah (Pesach vol 2 pg 42) concurs.
- There's a dispute in the Rishonim if one should lean during Karpas. Shibolei HaLeket (92b) writes that one shouldn't lean (probably for the reason that Karpas represents slavery and leaning represents freedom). However, Avudraham writes that one should lean during Karpas [perhaps the reasoning is that it's a mitzvah of the night and has some resemblance of freedom; see R. Moshe Shternbuch, Moadim U'Zmanim 7:183 who says that the Rambam would agree with the Avudraham]. The majority of Achronim including Mateh Moshe 625, Birkei Yosef 473:14, Shulchan Gavoha 473:29, Gra in Kiseh Eliyahu 473:4, and Chazon Ovadyah (Pesach vol 2 pg 40) rule like the Shibolei HaLeket and so one doesn't need to lean. Nonetheless, since by Maror is one wants one can lean (Bet Yosef 475, Magen Avraham, and Mishna Brurah 475:14), one is certainly allowed to lean during Karpas. See further elaboration at dailyhalacha.com.
- Mishna Brurah 473:55
- Chazon Ovadyah (Pesach vol 2 pg 35)