The laws of Passover as established at the time of the Exodus are:
- "do not have leaven in your habitations"
- "do not eat leaven"
- "have a celebration in honor of the Exodus -- in which you recount the Exodus to your children, eat Mazah (unleavened bread), "bitter herbs", and the sacrifice"
The sacrifice is forbidden without the Temple. So today the sacrifice is commemorated but not eaten.
The script for the celebration that we use today was finalized and formalized around two-thousand years ago.
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Traditionaly observant Jews
- clean thoroughly to eliminate or sell all items which may possible have the smallest trace of leaven (leaven is permitted to non-Jews - thus stuff can be sold)
- for the eight days of Passover only buy and eat items which are certified as "clean for Passover" - ie: they don't have any trace of leaven.
- keep the whole script of the celebration meal and may embelish on it -- add more details, more stories, more games...
Those who are not so observant may not eliminate all the leaven from the house,
they may or may not eat leavened products or create their own compromise structures where they don't eat bread but perhaps are not careful about less straight forward foods,
they may omit bits of the script of the celebration.
"Orthodox" is a generic classification in Western culture that generaly means the people are traditionaly observant
"Reform" is a movement which was started with the notion that Jewish observances are optional -- and therefore every reform home will do things differently
"Conservative" is a movement which separated from Reform with the notion to return to observance - but in practice many Conservative people are not traditionaly observant.
"Reconstructionist" was a movement started on the idea that the rituals are important for cultural and historical reasons but may or may not actualy believe in the underlying story. It would therefore be most likely that a Reconstructionist home will keep the historical celebration ritual but omit the detailed cleaning or guarantee of "clean for Passover"
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One of the other answers complicated things for you by including a discussion of "Kitniot" -
this is a series of items which certain local communities (most of the European ones) forbade by rabinic law as being inherently "uncleanable for Passover".
Today, all rabbis agree that the items may be obtained in a "clean for Passover" manner -- but people from those communities keep the law as a matter of cultural connection.
The choice to eat or not to eat Kitniot is not related to leve lof observance or the movement one belongs to. It is rather based on whether one adheres to this particular rabinic tradition.
Of course, if a person is not traditionaly observant, then certainly Kitniot is the first thing they tend to eliminate.