Learn Icelandic: Hann var að hjóla án hjálms. - He was cycling without a helmet.

The preposition án (without) is one of the special words that force the noun following it into the genitive case. For strong masculine nouns like hjálmur, the genitive singular is formed by replacing the -ur ending with an -s, making it hjálms.

Is the -s on hjálms the same as an English plural or possessive?

It looks like it, and it is actually historically related to the English possessive "s" (like in "a biker's helmet"). However, in this sentence, it is not showing possession or a plural (the plural is hjálmar). It is simply the grammatical case ending required because it comes right after án.

How does "var að hjóla" translate to "was cycling"?

Icelandic uses the verb vera (to be) + að + an infinitive verb to show an ongoing action. Since var is the past tense of "to be" for "he", var að hjóla specifically means he was currently in the middle of cycling, rather than just saying he "cycled".

Why isn't there a word for "a" before hjálms?

Icelandic does not have an indefinite article (a or an), so you just use the noun by itself. If the sentence instead meant "without the helmet", you would add the definite article suffix to the end of the genitive word, making it án hjálmsins.