These are three alternatives which all have the same meaning:
1. I shower
2. I take a shower
3. I have a shower
In the first sentence 'shower' is a verb. In the other two sentences, the word 'shower' is a noun: it's a singular, countable noun so it needs the article 'a'.
Generally speaking, where two forms exist, collocations with 'take' are more common in American English, while collocations with 'have' are more common in British English.