"Rain" is usually an uncountable noun, and "shower" is a typical way of making it countable. You could also say "rainstorm," although that implies more severe weather. http://dictionary.cambridge.org/dictionary/english/rain?q=Rain

Rain shower is a specific type of shower head that is usually big and produces enough water that makes you feel as if you are outdoors while it is raining - Mind you, I have never seen it raining that heavy before. Shower can also mean rain that only lasts for a short period of time. Example: We are expecting light showers later this evening. If it is snowing for a short period of time then you can say "snow shower".

阵雨 similarly, snow showers mean 阵雪

It means the rain is coming down like a shower. A lot of weathermen on TV will describe snow, rain, or hail with the word shower after it. It can mean that the rain is coming down overhead. It doesn't have anything to do with a shower for hygienic reasons. Hope this can help!