Traditional Church teaching identifies John the Apostle (one of the 12 apostles mentionned in the Gospels) with John the Evangelist (author of the Gospel of John) with John the author of the book of Revelations (John of Patmos) with the "disciple whom Jesus loved" (ὁ μαθητὴς ὃν ἠγάπα ὁ Ἰησοῦς). Whether this identification is accurate matters little to the answer of your question -- in any case, John was the name of possibly (along with Peter) one of the major characters of the Gospels in traditional Church teaching, and the quintessentially faithful disciple, omnipresent in Christian iconography as the one beardless disciple. Hence the importance of this first name.

Now looking first to this "disciple whom He loved" (Jn 21:20) do note that it is loved as in ἀγαπάω in the perfect form : the discipled he loved in a pure, divine way (in latin, agapè is translated into caritas, that is, etymologically, not charity but "whom he loved dearly or something akin) -- it is the same word used throughout the Gospel to discuss the Love God has for His people ; a deeper love that philia (brotherly/familial love) and a non carnal love (as opposed to eros).

As far as identification of John the Apostle with the Evangelist and John of Patmos, there is a definite link between the two last as a part of Johannine litterature, sharing several motifs, a general idea that Christianity stands in contrast rather than in continuity of ancient Judaism, and a more allegorical / mystical trend (to say this bluntly) -- but saying they are one person remains matter for debate. The Gospel of John is not considered to have been written first (it would be the Gospel of Mark, through intertextual analysis and consensus of exegets, around 70AD) making it unlikely John the Apostle would have lived that old to write it. It could though be linked to a Johannine community that recognized the Apostle as a moral authority and its source of teaching.

There is a very interesting and historically serious documentary series about all this (in French) called Corpus Christi ; with a sequel called les origines du christianisme and another called Apocalypse. I would recommend them.