A non-finite clause is the part of a sentence that typically functions as a dependent or subordinate clause within it and includes a verb in one of a few forms. This type of clause often includes of a verb in its infinitive form, which may require the auxiliary "to" with it. The suffixes "-ed" and "-ing" can also be used to change the verb within such a clause, though in all of these examples the verb is still functionally infinitive. This is where the name "non-finite clause" comes from, because the verb within it is not finite and does not express tense or aspect.
There are two major elements of a non-finite clause: it is a dependent or subordinate clause within a sentence, and the verb within it is non-finite. A clause is a large part of a sentence, and independent ones are essentially simple sentences that can stand on their own. Something like "He wrote a letter" is a simple sentence and is also an independent clause.
A dependent clause is similar in many ways to an independent one, except it lacks enough information to function on its own. It requires an independent clause, otherwise it is an incomplete sentence. For example, "to send to his mother" is a dependent clause since it lacks a subject and therefore is not a complete sentence. In this instance, it is a non-finite clause and might be joined by the previous independent one to form a full sentence as "He wrote a letter to send to his mother." This provides more information about the act of writing; in this case it is an adverbial that describes the purpose of the action.
The other major element of a non-finite clause is the form of the verb within it. Verbs in a non-finite clause are typically in the infinitive form, which often includes the auxiliary "to" with them, such as "to send" in the previous example. The infinitive form, which means the same as "non-finite," lacks tense or aspect. For example, in the sentence above, "wrote" is in the past tense and so it is finite, while "to send" is non-finite and does not express any particular tense.
A verb in a non-finite clause can also be a form other than the infinitive, typically with either an "-ed" or "-ing" suffix. For example, the sentence "Awakened from a deep sleep, he coughed violently," contains a non-finite clause in the form of "Awakened from a deep sleep." In this instance, the verb "awakened" is in the "-ed" form and is non-finite because it lacks a subject within the clause. The "-ing" suffix can also be used in much the same way, such as "Looking down from the stairs, the girl felt a sudden wave of vertigo." Both of these examples begin with dependent clauses that rely upon the independent ones that follow the commas.