Learn German: Wir schauen mal in die Zeitschrift. - We are just taking a look in the magazine.

Mal is a modal particle (a small "flavor" word) that makes an action sound casual, quick, or spontaneous. Without it, Wir schauen in die Zeitschrift sounds more like a serious, literal statement: "We are looking into the magazine." Adding mal softens it to a relaxed "We are just taking a look in the magazine."

Why do we use the accusative in die here instead of the dative in der? We aren't moving anywhere.

In is a two-way preposition. Even though you aren't physically walking anywhere, your gaze is being directed from the outside to the inside of the magazine. Because there is a "direction" or "destination" to your looking, German uses the accusative case (die Zeitschrift). You would only use the dative (der) to describe something that is already statically resting inside the magazine (like a photo).

How is a Zeitschrift different from a Zeitung?

A Zeitung is a newspaper (focused on daily news), while a Zeitschrift is a magazine or journal (often published weekly or monthly, literally translating to "time-writing"). Because they end in the classic feminine suffixes -ung and -schaft, they both use the article die!