
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.

shiver
A sudden feeling of cold, fear, or excitement that runs through the body; a chill or trembling.
Bei dem unheimlichen Geräusch lief ihm ein kalter Schauer über den Rücken.
At the eerie noise, a cold shiver ran down his spine.
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
Regen refers to precipitation in general and can continue for a longer time or fall over a wide area. A Schauer, by contrast, is typically brief, sudden, and limited to a particular area; it can also be heavier. Use Regen when talking about the weather in general, and Schauer when you mean a brief period of precipitation.

Am Nachmittag zog nur ein kurzer Schauer über den Ort.
In the afternoon, only a brief shower passed over the town.
Am Nachmittag fiel in der ganzen Stadt mehrere Stunden lang Regen.
In the afternoon, rain fell across the entire city for several hours.