der Schauer
shower
A short, usually sudden and sometimes heavy precipitation in the form of rain, hail, or snow.
Wir wurden auf dem Heimweg von einem heftigen Regenschauer überrascht.
We were surprised by a heavy rain shower on the way home.

shower / shiver
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shower
A short, usually sudden and sometimes heavy precipitation in the form of rain, hail, or snow.
Wir wurden auf dem Heimweg von einem heftigen Regenschauer überrascht.
We were surprised by a heavy rain shower on the way home.

The two meanings are easy to distinguish. If people are talking about the weather, rain, or clouds, the short precipitation is meant. If it is about feelings, fear, cold, or the body, the chill is meant.
Der Wetterbericht hat für den Nachmittag kurze Schauer angekündigt.
The weather report announced brief showers for the afternoon.
In a weather context, 'Schauer' is often made more precise by the type of precipitation: 'Regenschauer' (rain shower), 'Hagelschauer' (hail shower), or 'Schneeschauer' (snow shower). Adjectives such as 'kurz' (brief), 'plötzlich' (sudden), or 'heftig' (heavy) are also typical.
Ein heftiger Hagelschauer hat die Pflanzen im Garten beschädigt.
A heavy hail shower damaged the plants in the garden.
The meaning 'Frösteln' (shivering/chill) is almost always used in the fixed expression 'ein (kalter) Schauer lief jemandem (Dativ) über den Rücken' (a cold shiver ran down someone's spine). Pay attention to the dative for the person who has the feeling.
Als sie das Knarren hörte, lief ihr ein kalter Schauer über den Rücken.
When she heard the creaking, a cold shiver ran down her spine.
rain
'der Regen' and 'der Schauer' both describe precipitation. 'Regen' is the general word for water that falls from the sky in drops. It can be short or long, light or heavy. 'Schauer' means short, often sudden precipitation. A 'Schauer' can be rain, but also hail or snow. Learners may confuse the words because a rain shower looks like rain. The important difference is duration: 'Schauer' sounds short and sudden, while 'Regen' is more neutral and general.

Am Nachmittag kam ein kurzer Schauer, danach schien wieder die Sonne.
In the afternoon there was a short shower; after that, the sun shone again.
In der Stadt hört der Regen seit dem Morgen nicht auf.
In the city, the rain has not stopped since morning.