Assuming that
the babble of babies isn't an innate hyper-language (as the movie
Baby Geniuses suggests) this label applies literally to the first
language learned after that of one's parents (or
mother tongue) - but it frequently gets applied to any and all non-first languages.
Supposing, for instance, that you're a native east-Asian polyglot, conversant in Mandarin, Cantonese, Japanese and Korean - if you want to work, study or live in an English-speaking country, you might be expected to pick up some English-language training in an ESL (English as a Second Language) class; AFAIK, there are no ETL or EFL classes around 8)
The thing that I find interesting (enough to motivate me to write this node 8) about second languages is the way they're separated not just by time but by space. (Whaaat?) I'm not talking about the thousand-league distances between the countries or nations where the languages are usually spoken, but the subtle distinctions of centimetres and smaller between the physical areas in the human brain where the use of different languages is kept.
Languages learned during childhood live in an adjacent but different part of the brain from languages learned during later development (as long as they find at least occasional use - it's not uncommon for a child to lose the use of a first language altogether if they move early enough to a country where they find more cause to use the second language of their schoolmates than the first used only at home) - one reason there's an immediate, palpable and qualitative difference between children taught a second language in Early and Late Immersion situations. Even though there's often only as much as five years difference, time spent later simply is not as effective as time spent earlier - Late Immersion students may be able to pick up the grammar and vocabulary, but it'll take years to be able to speak accent-free, if ever.
But back to that space thing. It's been observed by numerous medical professionals that following trauma (eg. a stroke or damage caused removing a brain tumor) in or around Broca's area (the language centre of the brain) that occasionally the part of the brain where the damage occurred happened to be the location where the first language - but not necessarily language use in its entirety - lived. This manifests in the bewildering spectacle of the recovering victim permanently losing their first language, and having to rely entirely on their (frequently less-apt) second or later languages, which would be residing in a nearby and undamaged portion of the brain. (Of course, given that ~25% of brain tumors hit around Broca's area, the area where later language lives is as likely to be struck as the area where the first language lives - but so long as you've got a later language you're packing a spare 8)
Awkward as it might be, consider that the alternative would be living out the rest of your life without the ability for any language use! A more compelling reason to learn a second language after childhood I cannot conceive of.