Full Gospel is one of the names that denotes Pentecostalism. A lot of the earliest Pentecostal churches were called "Full Gospel Tabernacles." So, really, Full Gospel is a term like "Spirit-Filled." When someone says, I'm a Spirit-filled Christian, what they are saying is that they are part of the Pentecostal and charismatic movements, but for their own reasons they don't want to identify with any specific movement.
"Full Gospel" also relates to the four-fold or foursquare gospel of Pentecostalism: Jesus saves, Jesus baptizes with the Holy Spirit, Jesus heals, and Jesus is the soon coming King.