According to Chabad, the names of the Hebrew (Jewish) months are: Nissan, Iyar, Sivan, Tammuz, Menachem Av, Elul, Tishrei, Marcheshvan, Kislev, Tevet, Shevat and Adar. Judaism 101 abbreviates "Menachem Av" as simply "Av," spells "Tishrei" without the "e" ("Tishri"), and shortens "Marcheshvan" to "Cheshvan."
The Hebrew months are not the same as the months on the Western calendar (also known as the Civil calendar). Both Chabad and Judaism 101 give the approximate months on the Western calendar that correspond to the months on the Hebrew calendar: Nissan is in March-April; Iyar is in April-May; Sivan is in May-June; Tammuz is in June-July and Av is in July-August; Elul is in August-September; Tishri is in September-October and Cheshvan is in October-November. Furthermore, Kislev is in November-December; Tevet is in December-January; Shevat is in January-/February and Adar is in February-March.
It would not be correct to say that the Hebrew name for "March" is "Nissan" or that the English name for "Nissan" is "April." Nissan is a month on the Hebrew calendar; March and April are months on the Western calendar. One calendar can be laid over the other to determine the Hebrew month and the Western month in which a particular day occurs, but the two calendars remain distinct and separate.