The nature of variation is very important to an understanding of human linguistic ability in general: if human linguistic ability is very narrowly constrained by biological properties of the species, then languages must be very similar. If human linguistic ability is unconstrained, then languages might vary greatly.
But there are different ways to interpret similarities among languages. For example, the Latin language spoken by the Romans developed into Spanish in Spain and Italian in Italy. Similarities between Spanish and Italian are in many cases due to both being descended from Latin. So in principle, if two languages share some property, this property might either be due to common inheritance or due to some property of the human language faculty.
Often, the possibility of common inheritance can be essentially ruled out. Given the fact that learning language comes quite easily to humans, it can be assumed that languages have been spoken at least as long as there have been biologically modern humans, probably at least fifty thousand years. Independent measures of language change (for example, comparing the language of ancient texts to the daughter languages spoken today) suggest that change is rapid enough to make it impossible to reconstruct a language that was spoken so long ago; as a consequence of this, common features of languages spoken in different parts of the world are not normally taken as evidence for common ancestry.
Even more striking, there are documented cases of sign languages being developed in communities of congenitally deaf people who could not have been exposed to spoken language. The properties of these sign languages have been shown to conform generally to many of the properties of spoken languages, strengthening the hypothesis that those properties are not due to common ancestry but to more general characteristics of the way languages are learned.
Loosely speaking, the collection of properties which all languages share can be referred to as “universal grammar” (or UG). However, there is much debate around this topic and the term is used in several different ways.
Universal properties of language may be partly due to universal aspects of human experience; for example all humans experience water, and the fact that all human languages have a word for water is probably not unrelated to this fact. The challenging questions regarding universal grammar generally require one to control this factor. Clearly, experience is part of the process by which individuals learn languages. But experience by itself is not enough, since animals raised around people learn extremely little human language, if any at all.
A more interesting example is this: suppose that all human languages distinguish nouns from verbs (this is generally believed to be true). This would require a more sophisticated explanation, since nouns and verbs do not exist in the world, apart from languages that make use of them.
In general, a property of UG could be due to general properties of human cognition, or due to some property of human cognition that is specific to language. Too little is understood about human cognition in general to allow a meaningful distinction to be made. As a result, generalizations are often stated in theoretical linguistics without a stand being taken on whether the generalization could have some bearing on other aspects of cognition.