For which species must we wait until after pesach to eat the previous crop? Are there different cut-off dates for different species?
This only applies to the five grains: wheat, barley, oats, spelt, and rye. They all have the same cutoff date (16th of Nisan).
חמשה דברים חיבין בחלה, החטים והשעורים והכסמין ושבלת שועל ושיפון. הרי אלו חיבין בחלה, ומצטרפין זה עם זה, ואסורין בחדש מלפני הפסח, ומלקצור מלפני העמר. ואם השרישו קדם לעמר, העמר מתירן. ואם לאו, אסורין עד שיבוא העמר הבא.
Five [species] are obligated in Challah: chittah, se'orah, kusemet, shibbolet shu'al, and shifon. These are obligated in Challah and they combine with each other [to form the requisite minimum amount of dough], and they are prohibited as Chodosh before Pesach and to reap [them] before the Omer [offering is brought]. If they took root before the Omer [offering is brought], the Omer [offering] permits them. If not, they remain forbidden until the next Omer [offering] is brough. (my translation)
That's the first Mishna in tractate Challah. In short, the Chadash prohibition applies to those same "5 grains" that can become chametz and that you say mezonot/hamotzi on (among other rules). The cutoff for all of them is the day of the offering of the Korban HaOmer, namely 16 Nissan. (In the Diaspora due to doubt the date is 17 Nissan but practically this is irrelevant as any Kosher for Pesach food you'll find nowadays is already certified Yoshon.)
Translating the names of the species is somewhat controversial as is defining exactly what cultivars are included in the same species. Roughly speaking, we're talking about the members of Triticeae, though a few Rishonim included members of Avena as well.
The Rabbis taught that the prohibition of חדש (new grain) only applies to the five types of דגן (grain) that undergo חימוץ (fermentation): חטים, שעורים, כסמין, שבלת שועל, שיפון. Traditionally, these have been assumed to be wheat and barley which are mentioned explicitly in the Torah ("ארץ חטה ושעורה"), and spelt, oats, and rye, which the Rabbis said are subsumed in חטה ושעורה. Of the 5, oats is the one most controversially included considering its negligible gluten content (see e.g. http://www.crcweb.org/kosher_articles/Celiac%20-%20A%20Guide%20to%20Halachic%20Observance%20%28JoHaCS%202010%29.pdf esp. fn 5). The biblical cutoff date is Nissan 16 or in the Diaspora, Nissan 17. In addition, J. Herman in Monsey, NY publishes annual "Chodosh Guides" that indicate (based on USDA crop reports) cutoff dates for the various crops and grain-based consumer products before which one can be relatively certain chodosh has not yet hit store shelves. These vary from crop to crop based on different harvest dates. (See, e.g., http://www.crcweb.org/yoshon_guide_prelim_5775.pdf )