Use the second person, which includes the personal pronouns "he," "she," "him," "her," "they" and "them," as well as proper nouns, such as names of authors and titles of publications. If it helps, keep a list of personal pronouns at hand to jog your memory when trying to rephrase a sentence.
Use "this author" or "this writer" to refer to yourself only when absolutely necessary. These often sound forced in everyday English but are permissible in formal writing.
Refer to the essay or parts of the essay, not to yourself. For example, write, "This essay will describe the history of sculpture," not "I will describe the history of sculpture" and "Part three will critique Hume's philosophy," not "I will critique Hume's philosophy in part three."
Make statements directly instead of explaining that they reflect your own thoughts. When you state something and support it with evidence, the reader can take for granted that you are sharing your own views. Write "X and Y show Z," not "I believe that X and Y show Z."
Find sources that make your argument or point for you, and cite them in your essay instead of referring to yourself. So, for example, write, "Smith argues that ..." instead of "I think that ..."
Use the passive voice to describe activities or events involving you. For example, instead of "I carried out the research," write, "The research was carried out." Rather than "I gathered results," write, "Results were gathered."