Any party who files, or defends a claim in a court of law must set out the facts which support the claim or the defence. Lawyers often refer to these as allegations of fact, assertions of truth or, more technically, pleadings or averments.
Mozley and Whiteley's Law Dictionary:
"A positive statement of facts as opposed to an argumentative or inferential one."
It is traditional for the courts to demand averments which are succinct, written simply and to the point, and not in literary, flowery or poetic style.
The document(s) which a party to litigation uses to set out its position may also contain argument, which purports to mix the allegations of fact (averments) with the statements of what the applicable law is. Thus, the word averment or averments is used to distinguish the allegations of fact against allegations of persuasive argument in law.
Averments is an older term of the common law, taken from legal dictionaries several centuries old but still in use in certain jurisdictions where plain language law is not widely celebrated by bench and bar. Notably, attorneys and courts in the United States routinely refer to averments to refer to the pleadings, in form or in content, of a party to litigation.