Learn German: Ich brauche heute einen Pullover.

Pullover is a masculine noun (der Pullover). In this sentence, it is the direct object—the thing being needed—which means it takes the accusative case. For masculine nouns in the accusative case, the indefinite article ein (a) changes to einen.

Why is heute (today) placed before einen Pullover?

In English, we usually put time words like "today" at the very end of the sentence. In German, time words typically come right after the conjugated verb and before the object. That is why heute sits between brauche and einen Pullover.

How would I change this sentence to say "I don't need a sweater today"?

You would replace einen with the negative article keinen. Just like ein changes to einen for a masculine object, the negative word kein (not a / no) changes to keinen. The sentence becomes: Ich brauche heute keinen Pullover.

Is brauchen (to need) a regular verb?

Yes, it is completely regular. To match the subject ich (I), you simply drop the -en ending from brauchen and add an -e, giving you brauche.