That sentence indicates one of the differences between Spanish and English. Although English requires the word "a" before "mariner" and "captain," Spanish doesn't require an equivalent word, which in this case would be un (if sung by a male).
The words "a" and "an" are known to grammarians as indefinite articles, and the Spanish equivalents are un (used before masculine nouns and noun phrases) and una (feminine). Using the Spanish indefinite articles when they aren't needed is one of the pitfalls for many beginning Spanish students. Say no soy un marinero, soy un capitán, and it would sound as awkward (and improper) as one possible translation into English: "I am not one mariner, I am one captain."
Generally speaking, whenever you use un or una in Spanish, you need to use "a" or "an" to say the equivalent in English. But the reverse isn't true. The appearance is that Spanish frequently "omits" the indefinite articles. Although the rules of article usage can get complex, and there are cases where the use of un/una is discretionary, here are the most common cases where the article shouldn't be used even though it's used in English:
Before an unmodified noun after a form of ser ("to be"), especially in reference to occupation, religion, affiliation or social status (normally, if the noun is modified, the article should be used): Soy profesor. I am a teacher. Él es un buen dentista. He is a good dentist. ¿Eres católica? —No, soy una metodista feliz. "Are you a Catholic." "No, I'm a happy Methodist." Es artista. She is an artist. Es una artista que muere de hambre. She is a starving artist.
Before otro ("other"): Quisiera otra taza. I would like another cup. Compró otro coche. He bought another car.
Before mil ("thousand") and cien ("hundred"): Gana mil dólares por mes. He earns a thousand dollars per month. Tiene cien años. She is a hundred years old.
In exclamations using qué ("what"): ¡Qué lástima! What a shame! ¡Qué casa! What a house!
After con ("with") and sin ("without"): Come con cuchara. She eats with a spoon. Escribe sin ordenador. He writes without a computer.
Frequently after forms of tener ("to have"), comprar ("to buy"), llevar ("to wear") and some other verbs when generically referring to things that people would normally have or use one at a time: No tengo coche. I don't have a car. Lleva camisa. He is wearing a shirt. Vamos a comprar casa. We're going to buy a house. ¿Tiene madre? Does he have a mother?
If you're a beginning Spanish student, don't sweat the details of the final rule for now. As you use the language, you will get a feel for whether you need the article or not.
Finally, there is one case where we don't use the indefinite article in English where it's needed in Spanish: in a series of two or more words joined by "and" (y in Spanish). In English we might say "a cat and dog," but in Spanish it must be un gato y un perro. Without the second un, the phrase would be understood as referring to one creature, a cross between a cat and dog. Note the distinction in these sentences: Conozco a un artista y un dentista means "I know an artist and I know a dentist," while Conozco a un artista y dentista means "I know a dentist who is also an artist."