Learn Hardcore German: Sie läuft so schnell wie ihr Bruder.

It’s the standard way to express equality (“as … as”). The structure is: so + adjective/adverb + wie + comparator. In the sentence, that’s so schnell wie ihr Bruder. Other examples: Er ist so groß wie ich. / Wir arbeiten so hart wie sie.

Why is it wie here and not als?

Use wie for equality (as … as) and als for inequality (comparative with “than”).

Equality: Sie läuft so schnell wie ihr Bruder.
Inequality: Sie läuft schneller als ihr Bruder.

Is so mandatory? Can I say “schnell wie ihr Bruder”?

In standard German, include so. Sie läuft schnell wie ihr Bruder is colloquial/regional. You can also strengthen equality with genauso or ebenso: Sie läuft genauso/ebenso schnell wie ihr Bruder.

What case is ihr Bruder, and why not ihren Bruder?

Ihr Bruder is nominative. After wie, the case depends on the role in the (often omitted) clause. Here it stands for the subject of an implied verb: … wie ihr Bruder (läuft), so nominative. Ihren Bruder would be accusative (object) and is wrong here.

Why is it ihr Bruder and not ihre Bruder?

Because Bruder is masculine nominative singular. Possessive determiners (mein/dein/sein/ihr …) behave like “ein-words”: no ending in masculine nominative. Compare:

Masculine nominative: ihr Bruder
Masculine accusative: ihren Bruder
Feminine (nom/acc): ihre Schwester
Plural (nom/acc): ihre Brüder

What does ihr mean here? Could it also mean “their” or “your”?

Lowercase ihr can mean her or their; context decides. Capitalized Ihr means your (formal). In our sentence it’s lowercase, so “her” (most likely, given the subject Sie = “she”), though “their” is possible in other contexts.

How do I know Sie means “she” and not “they” or formal “you”?

From the verb form. Läuft is 3rd person singular, so Sie läuft = “she runs.” “They” or formal “you” would take plural: Sie laufen.

Is schnell an adjective or an adverb here? Do I need an ending?

Here it’s an adverb modifying the verb läuft, so no ending: Sie läuft schnell.
As an attributive adjective before a noun, it takes endings: ein schneller Läufer. As a predicate adjective with “to be,” no ending: Sie ist schnell.

What is the infinitive of läuft, and why does the vowel change?

The infinitive is laufen. It’s a strong verb with a stem vowel change (au → äu) in the 2nd and 3rd person singular present:

ich laufe, du läufst, er/sie/es läuft, wir laufen, ihr lauft, sie/Sie laufen.

Can I move parts of the sentence for emphasis?

Yes. You can front the comparison phrase; the verb stays in second position: So schnell wie ihr Bruder läuft sie. You can also add focus words: Sie läuft genauso schnell wie ihr Bruder.

Can I expand the wie-part into a full clause, and do I need a comma?

Yes. If you use a full clause after wie, you need a comma: Sie läuft so schnell, wie ihr Bruder läuft. If you omit the repeated verb, no comma: Sie läuft so schnell wie ihr Bruder.

Does laufen mean “to walk” or “to run”?

In standard German (especially in Germany), laufen typically means “to run.” In some regions and in casual speech, it can also mean “to go on foot” (“walk”). Gehen is the default verb for “to walk.”

How do you pronounce läuft?

äu is pronounced like the “oy” in English “boy,” so läuft ≈ “loyft.” The final -ft is pronounced clearly. Bruder has a long u: “BROO-der.”

Is there any difference between so schnell wie, genauso schnell wie, and ebenso schnell wie? What about gleich schnell?

so schnell wie is the neutral, default equality pattern.
genauso schnell wie adds “exactly/as much as” emphasis.
ebenso schnell wie is a stylistic synonym of “genauso.”
gleich schnell (without wie) is also fine: Sie sind gleich schnell. The form gleich schnell wie is common regionally (e.g., in Switzerland/Austria) but is less standard in Germany; prefer so/genauso/ebenso … wie there.