Learn German: Fehlt hier irgendetwas? - Is something missing here?

etwas simply means "something". When you add irgend- to the beginning of a word, it adds a feeling of "any at all" or randomness. So irgendetwas translates well to "anything" or "anything at all".

Is irgendetwas singular or plural? Why is the verb fehlt?

irgendetwas is always treated as a singular concept, just like "something" or "anything" in English. Because of this, the verb takes the singular "it" form: fehlt (is missing).

I often hear Germans say irgendwas. Is that the same as irgendetwas?

Yes! In casual, spoken German, etwas is very often shortened to just was. Therefore, irgendetwas becomes irgendwas. They mean exactly the same thing and can be used interchangeably in conversation.

How exactly does the verb fehlen work here? Can I use it to say I miss a person?

fehlen means "to be missing" or "to lack". In this sentence, the thing that is missing (irgendetwas) is the subject. You can use it for people, but the grammar flips! To say "I miss you", you say Du fehlst mir (literally: "You are missing to me").