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?