There is no single answer to this question. More than movement, potential conversion students should instead focus on which rabbi they feel they connect with. You can view our rabbis here.
Below are some basic explanations of what the movements are and what that means for your conversion process.
Orthodox Judaism focuses its religious practices around halacha, Jewish law. But Orthodox Judaism is not just legalism. Orthodoxy is filled with a diversity of religious approaches and focuses, from the rationalistic to mystical. Orthodox rabbis will require observing kashrut, Shabbat, holidays, family purity laws and participating in local Orthodox communities. Men must be circumcised in accordance with Orthodox standards. Orthodox rabbis will not offer conversion if it creates an interfaith relationship/marriage.
Conservative (called Masorti outside of the United States) seeks to preserve Jewish tradition and ritual but has a more flexible approach to the interpretation of the law than Orthodox Judaism. Conservative rabbis, while theologically diverse, will nonetheless expect students to traditionally observe kashrut, Shabbat, and participate in local Jewish communities (synagogues). Conservative rabbis will accept non-Conservative conversions provided that mikvah and brit milah (or hatafat dam brit) were part of the conversion.
Reform Judaism is the first progressive Jewish movement. Its focus has been primarily on the spiritual and ethical/moral aspects of Judaism instead of the ritual or ‘legal’ parts of traditional Judaism. Reform rabbis are generally flexible on religious practice in favor of social justice/tikkun olam. Reform Judaism accepts the conversions of all other movements.
Jewish Renewal is a recent movement in Judaism which endeavors to reinvigorate modern Judaism with Kabbalistic, Hasidic, and musical practices. Renewal reintroduces the "ancient Judaic traditions of mysticism and meditation, gender equality and ecstatic prayer" to synagogue services. There is no standard for accepting conversions of other movements, but generally speaking Renewal rabbis will accept any conversion from Orthodox or non-Orthodox rabbis.
Humanistic Judaism was founded in the 1960's by Rabbi Sherwin Wine and is a non-theistic Jewish alternative that places the emphasis on humanism through the lens of Jewish identity. Judaism is not the religion of the Jewish people, but rather the culture and experience of the Jewish people. Non-theistic prayers/meditations substitute traditional prayer and Humanistic Jews tend to focus on things like Jewish education, history and social action as expressions of Jewish values.
Post-Denominational and Pluralistic are not the names of Jewish movements. These are labels that rabbis (and lay Jews) may use to signify that they are not part of a movement (such as Reform or Conservative). Post-Denominational and Pluralistic rabbis draw from different movements and may be very traditional or non-traditional. These rabbis have ordination from non-movement seminaries but are nonetheless valid rabbis.
If you are seeking recognition as a Jew from one of these movements, you should convert with that movement.