Higher criticism is a Christian theological doctrine and school that described itself as passing beyond what its practitioners saw as the preparatory stage of textual criticism to look at the Bible critically. "Higher" in this sense is much like the "meta" in "metaphysics" (see "the origin of the word 'metaphysics').
[The above seems to be an attack on higher criticism, rather than an explanation of it.]
Lower textual criticism tries to establish the most accurate version of a text. Higher textual criticism tries to determine the authorship of a text. The primary goal of both fields of research is the search for historical truth. For some, however, the goal of higher textual criticism of the Bible was to demythologize the Bible.
Traditional Christians and Orthodox Jews contend that higher criticism assumes what it attempts to show, that without the miracles, the Bible lacks evidence of a Deity. Scholars respond by pointing out that belief in God has nothing to do with belief in whether a certain text has more than one author.