What Is Language?

What is (a) language a property of? What does (a) language belong to?

To the individual?
The language exists in the brain of the individual.
To the producer/perceiver pair?
The language can only be described in terms of the interaction between a speaker (writer, signer) and a hearer (reader, sign observer).
To the community?
The language is a system that is shared by the members of a community, evolving over time as the community evolves.

Within an individual?
- All language users know a spoken or signed language. Many also know a written version of a spoken language.
- Most (all?) speakers control more than one register of a language.
- Many people know more than one dialect of a language or more than one language.
- All language users make errors sometimes.
Within a community?
- Each person has an idiolect different from all others.
- Different geographical areas are characterized by different regional dialects.
- Different social groups within the same geographical area may have different social dialects.
- Particular dialects may have more prestige than others, but apparently never for purely linguistic reasons.
- Standard dialects may be sanctioned by political entities.
- Particular forms or rules may be prescribed by educational systems or social commentators (but never linguists), even when they are not actually used in natural speech.

It is (at least in part) learned. This is true of all levels of language. Thus quite different languages are used by a single species, and languages change over (historical) time.
It is characterized by at least two distinct formal levels, phonology and morphosyntax, each with its own units and regularity.
New utterances may be created by recombining familiar words, and the number of possible utterances is apparently infinite. Thus the human language capacity includes the ability to comprehend and produce novel utterances. Human language is productive and systematic.
Utterances can be "about" objects and situations which are not "there" at the time of the utterance. In fact utterances can be about imaginary, even impossible situations.

They are learned. In fact the stages in learning are very similar to those for spoken languages.
Signed languages are characterized by the same levels as spoken languages. In fact they resemble spoken languages in most of the important details within each level.
Signed languages are productive and systematic.
Signed languages have the capacity to refer to the same range of situations as spoken languages, including past, tuture, and impossible ones.
Signed languages also differ from spoken languages in interesting ways, in particular in their degree of iconicity.

Two languages may be genetically related; that is, they are similar because they share a common ancestral language. For example, Spanish, Friench, and Italian are related because they both descend from Latin. Spanish, English, Russian, and Hindi are related because they all descend from a (reconstructed) language, Proto-Indo-European.
Two languages may resemble each other because they have been in contact for long periods of time. For example, Japanese has many words of Chinese origin even though it is probably unrelated to Chinese genetically. Hindi and Tamil share many phonological properties even though they are probably unrelated genetically. Because languages may borrow from each other, it may be difficult to establish genetic relationships on the basis of similarity.
Two languages may resemble each other because they are typologically similar. That is, there may be combinations of properties which naturally go together in a language. For example, Japanese, Hindi, and Turkish are all languages in which the verb is normally final in sentences, and these languages share a number of properties even though they are probably genetically unrelated.