Learn Polish: Musiałem obejść tę dużą kałużę. - I had to walk around this big puddle.

It is formed by adding the prefix obe- (a variation of ob-, which means "around" or "bypassing") to a form of the base verb iść (to walk/go on foot). This creates a new perfective verb meaning to physically walk around an obstacle to avoid it.

Why does the prefix have an 'e' at the end (obe-) instead of just being ob-?

The prefix ob- often gains an extra -e- when attached to a root starting with a tricky consonant cluster, like -jść in obejść. It is simply there to make the word easier to pronounce. You will see this same pattern in other verbs, like obejrzeć (to watch/inspect).

What case is used for "tę dużą kałużę" and why?

It is in the Accusative case. Even though in English we say "walk around a puddle" (which sounds like it might need a preposition), verbs of motion with the ob- or obe- prefix take a direct object in Polish. You are directly "bypassing" the puddle, so the feminine noun kałuża and its modifiers take their Accusative endings.

I often hear native speakers say "tą" instead of "tę" in the Accusative. Why is it "tę dużą kałużę" here?

You spotted a very common exception! The strictly correct Accusative form of the pronoun ta is tę. However, because feminine adjectives in the Accusative end in -ą (like dużą), many native speakers in casual conversation will say tą dużą kałużę to make it rhyme. While common in speech, tę is what you should use in writing and careful speech.