The first person in grammar is the one making a statement, the speaker or writer - in contrast to the person (or persons) being addressed, i.e., spoken or written to (the second person), and the person (or persons) being spoken or written about (the third person). In English the first person pronouns are 'I' and 'me' in the singular; and 'we' and 'us' in the plural. For more see person.
Avoid the first person in academic writing. Should you notice that this guide does not always do this, remember to do as I say and not as I do.
This advice also varies among disciplines. In mathematics, for example, it is customary to use the first person plural: "Applying the corollary, we can see that..." This is called the authorial 'we'.
For a note on a particular grammatical problem in writing formal English, you may want to see Agreement of preposition with pronoun.