Typically, developing children acquire the language that is spoken and heard in their homes, making that their first language (or L1). In a home where only English is spoken and heard, English will become the L1. In a home where Spanish and German are spoken and heard equally, a child will develop two L1s--both Spanish and German. A first language (again, for typically developing children) will emerge without any direct instruction. Children will certainly benefit from good modeling and direct conversations, but children have an innate ability to acquire the language they are immersed in. For this first language to develop well, it is important that children be given adequate exposure to things like correct grammar, word order, semantics, a wide vocabulary, and inflections.
Both children and adults have the capacity to also learn a second language (or L2). Unlike the first language, which solidifies during a window until about age 12, the L2 is actively studied and practiced. New information must be rerouted and reprocessed in the brain as L2 learners meld the language patterns they already know with those of the new language. For example, a student who has learned English as an L1 has absorbed a rule that dictates an adjective preceding a noun in typical speech. As they develop an L2 in Spanish, they must rewire this, consciously remembering to place adjectives after nouns instead.
The fundamental difference is in how the language information is solidified. While the L1 takes almost no effort besides immersion in the language, the acquisition of an L2 is very deliberate.