Rabbi Asher ben Yechiel (Rosh), is one of the most crucial Rishonim on the Talmud. He is printed in the back of almost every Talmud, and is useful both as a halachacist, and also as one of the last tosafists he often clarifies and explains their position.
He is the bridge between Talmudic commentary and Halachic works. His son, the Tur (Yaakov ben Asher) is the author of the Arba Turim, which is the predecessor of the Shulchan Aruch. The Tur also wrote ‘piskei harosh’ which is a summary of the halacha derived from the Rosh’s Talmudic commentary and is also printed in the back of most editions of the Talmud.
The Rosh was also the bridge between the Ashkenazi and Sefardi worlds, forced as he was to resettle and move from Northern Europe to Spain. His commentary is therefore an interesting commentary on the different customs of East and West at the time. R’ Yosef Karo uses him as the ‘Ashkenazi’ amonst his three main sources (the other two being the Rif and Rambam).
Here is a shiur I gave about the Rosh for WebYeshiva
May His Soul be Bound in the Bonds of Eternal Life