Learn Italian: Lei fa una doccia veloce. - She takes a quick shower.

In Italian, you don't "take" a shower, you "make" or "do" one! The verb fare is used in many common expressions where English uses a different verb, like taking a shower (fare una doccia) or taking a trip (fare un viaggio).

How do I know who is taking the shower just from the word fa?

The verb form fa is the "he/she/it" form (third-person singular) of the irregular verb fare. In this sentence, the word Lei (she) clarifies exactly who is doing it, though Italians will often drop Lei if it is already obvious from the context.

Why is the adjective veloce placed after una doccia?

In Italian, most descriptive adjectives come right after the noun they describe. So instead of "a quick shower", you say "a shower quick" (una doccia veloce).

Una doccia is feminine. Why doesn't veloce end in an -a to match it?

Adjectives that end in -e in their dictionary form, like veloce or intelligente, are used for both masculine and feminine nouns. They do not change for gender, only for number (changing to -i when plural).