Here's an attempt to layout the history of the position, and the time periods in which it changed. All of this information can be found here, although it's buried with all of the history of every other position as well. This question requires a good understanding of nomenclature, and I would encourage you to read the entire article in the link, especially the sections labelled "Age of Confusion" and "Making Sense of it All." As the article states:
The trend away from geometric naming of the offensive positions has led to considerable confusion
So here are the bullet points that help identify the evolution of the fullback position. The origin of the fullback was in 19th century rugby. Originally these positions were named "tends," but the evolution of the game created some new positions with new names.
As the game became more sophisticated, backs positioned at different depths (i.e. distances behind the forwards) were further differentiated into half back, three quarters (the fraction 3/4) back, and full back... It was the Irish nomenclature of quarter back, half back, and full back that came to North America for use in what was to become the dominant native form of football. The terms became hyphenated and eventually unhyphenated single words, "quarterback" (QB), "halfback" (HB), and "fullback" (FB).
In the early days of American football (around the beginning of the 20th century), the role of the fullback was altered. Instead of the position just indicating a position behind the line of scrimmage, it began to also identify the role and physical nature of the position.
Over time, the typically fast back who played fullback, on offense and defense, was replaced by a heavier one who presented a greater threat to run with the ball more or less straight ahead, and to tackle his opponent trying to do the same. For this purpose, the player tended to be placed closer to the line than previously—often as far forward as the halfbacks (the offensive backs then forming a letter T, sometimes called a "straight T", later to distinguish it from slight variants wherein other backs did not form a line perpendicular to that from the quarterback) or even farther forward. But the position kept being called "fullback".
In the 1960s, many non-professional teams lost the distinction between a halfback and fullback. The NFL maintained the nomenclature of halfback and fullback:
...the distinction between fullback and halfback was erased, each replaced by "running back" (RB). That term became popular during the 1960s as well, although even into the 1970s, some playing the pro set kept the HB-FB distinction... The distinction referred to the build of the players, the fullback being a stronger runner more or less straight ahead, and the halfback faster to attack the defense's flanks. However, on many teams there was no strong distinction between those backs, and since their major role was running with the ball, "running back" was most descriptive without maintaining a spurious geometric connotation that'd become outmoded by their lining up in different depth relationships.
This trend has continued on. The present day explanation of fullbacks:
[T]o this day many teams maintain the offense's fullback-halfback distinction... [S]ome formations have placed the heavy back even more starkly forward, to function as a blocker for the running backs.
There are a number of current NFL teams that do not have a full-time fullback in their starting lineup. The three with no fullback listed:
- New England Patriots
- Indianapolis Colts
- Detroit Lions
The four that have a hybrid fullback:
- Kansas City Chiefs (Peyton Hillis is listed as a RB/FB)
- Denver Broncos (Chris Gronkowski is listed as FB/TE)
- Chicago Bears (Evan Rodriguez is listed as a FB/TE)
- Seattle Seahawks (Michael Robinson listed as RB/FB)
In some instances, teams will line up a defensive lineman as a fullback:
In selected plays, some teams will have a defensive lineman report as an eligible receiver to line up as a fullback. Examples of such players who have been frequently used as situational fullbacks include Haloti Ngata, Vince Wilfork, and Isaac Sopoaga.