This is a Hebrew term used with reference to Pesach. Literally means pulses or legumes.
On Pesach, Orthodox Jews won't eat any Chametz. Not only this, they won't even own any Chametz for the duration of the festival, with all chametz in their house being nominally sold to a non-Jew for the duration of the festival, and bought back (usually by the local Rabbi on behalf of the whole community) afterwards. The prohibitions against eating Chametz are incredibly strong it is better for a Jew to eat pork the whole year round than eat a single slice of bread on Pesach.
However, there is another category of food that Ashkenazi Jews (those of Eastern European descent) don't eat. This is Kitniyot. This includes rice, corn, soy beans, string beans, peas, lentils, mustard, sesame seeds and poppy seeds. Why are these not eaten even though they're not Chametz?
The given reason is that their products appear like chametz products. For example, it can be hard to distinguish between rice flour wheat flour (which would be chametz). Therefore, to prevent confusion, all kitniyot was prohibited. Another Rabbi also notes that due to crop cycles, chametz and kitniyot grains may get mixed together, therefore causing one to inadvertently eat chametz.
However, the restrictions on Kitniyot are much less strict than those on Chametz (see above). It is permitted to own Kitniyot on Pesach, and even use it for non-food purposes (eg some baby powder contains cornstarch). Furthermore, one is permitted to continue taking medicines that contain Kitniyot on Pesach, whereas a medicine containing Chametz could only be taken if it was a true matter of life and death.
As mentioned above, this all applies to Ashkenazi Jews, but not the Sephardi Jews. This creates the strange situation, for example, where a Sefardi family could be eating rice on Passover -- whereas their Ashkenazi neighbors will not! Furthermore, the Ashkenazi family could go to the Sephardi family on Pesach, and be served a plate of (say) meat, potatoes and rice. Although they shouldn't eat the rice, they don't have to refuse the entire plate.
What about derivatives of kitniyot -- e.g. corn oil, peanut oil, etc? This is a difference of opinion. Many will use kitniyot-based oils on Passover, while others are strict to only use olive or walnut oil.