@Gehrman a
tafsir is basically a commentary on the Qur'an which seeks to place the surahs in context and deduce broader meaning from them. As such it is an extremely important genre in the (ongoing) history of the production of Islamic law. However different commentators will offer different interpretations of the verses based on their particular approach to a variety of matters. To give three famous examples - the
tafsir of Ibn Kathir, a 14th century Shafi'i scholar, is I believe regarded as one of if not the most authoritative works in the genre by Sunni Muslims all the way to the modern age. You will find that later scholars will regularly refer back to his work a lot, even in cases where they may differ slightly. It is said to reflect a fairly straightforward, literal approach to the text, avoiding metaphorical speculation and such. Another famous example is Sayid Qutb's
In the Shade of the Qur'an. Qutb was an important figure in the Egyptian Muslim Brotherhood in the 40s-60s and his legacy helped shape the thoughtof Islamist and jihadist groups long after his death. So you might expect his
tafsir to reflect the Islamist outlook of the Islamist trend, with an emphasis on the Qur'an's revolutionary potential. Finally there is Muhammad Asad's
Meaning of the Qur'an, which includes his own English translation alongside his commentary. Asad was an Austrian-Jewish convert to Islam in the early 20th century, who, among other things, befriended and advised Ibn Sa'ud and later became a citizen and diplomat in the new state of Pakistan (he also wrote a really interesting memoir called
The Road to Mecca which I always recommend as a great examination of the spiritual appeal of Islam for converts). His approach would reflect what is called Islamic Modernism, where the concern would be to reconcile the meaning of the text with the modern world. Of the three, I would guess Asad's is likely the least popular among Muslims today. It should be noted that Ibn Kathir is the only one of the three who would be recognized as an 'alim (scholar), the other two being laymen.
I mention these three not only because they are three of the most famous examples of the genre representing three distinct trends, but because they're all also easily available to find online in English translation. So you will have no problem dipping into them and finding out their approach to these surahs.