Is there a word to describe the kind of sentence where one uses one's own point to prove one's point?

This type of argumentation is called "circular reasoning" or "tautological".

When the circularity of the argument is less blatant, and the tautology is indirect or merely implied by one of the premises, then the fallacy is known as "begging the question".

In recent years, "begging the question" has started to be used as a synonym for "raising the question", so if you wanted to avoid ambiguity, or use the term in a more formal context (e.g. philosophy, rhetoric), you could use one of the synonyms "assuming the consequent" or "petitio principii".

Circular reasoning is really the correct term. Technically, this is also "begging the question," but as was previously pointed out, common (mis)use of that term has introduced ambiguity. Although the term "tautology" is often used to describe this type of situation, tautologies and circular reasoning are not necessarily the same thing. A tautology is a statement which is true by definition, including those which do not involve circular reasoning. A tautology may not be a logical flaw. For instance, it is common practice for mathematical proofs to begin include some tautologies among the postulates that are used to prove the theory.