A liturgical rite is a way to celebrate the Mass. At least for the purpose of this question I will assume this meaning of the term.
The standard way Mass is celebrated by Roman Catholics is in the Roman Rite. This normally refers to celebration of the Mass originally based on the Missale Romanum from 1970 as commissioned by Pope Paul VI (there have been recent modifications); this happens (always?) in the local language. The so-called extraordinary form of the Roman Rite is the Tridentine Mass, celebrated in Latin, nowadays according to the Missale Romanum of 1962.
Aside from these two liturgical rites (which some will say are technically just two forms of a single rite), I have seen mention of others, such as the Ambrosian Rite, Anglican Use, and Zaire Use. Which forms of Mass (rites) are presently used within the Roman Catholic Church? How can one characterize them, and where are they used?