I have been rather confused about this for a while, mainly when someone says they will play their Champion as a carry when the Champion is not classified as one. So, I suppose I am also wondering the definition of fighter too and what makes a carry a carry, more than just the definition.
From the League of Legends Wiki:
"Carry" is a term that is typically used to refer to the primary damage dealer / killer on a team. So you will usually see a carry have the highest number of kills on that team, by the end of the game. Carrys may often start with low effectiveness at early game and rise to power through the mid / late game. The carry is usually a character that your team wants to feed, you want him to farm and get the best gear possible so he can own.
Basically, a carry starts out slow but becomes the most powerful member of the team if played effectively. If you feed a carry, his eventual power is designed to be greater than it would have been if you'd fed a fighter instead.
Fighters are champions that have constant damage output and good survivability. They tend to last a little longer in team fights as opposed to an assassin, deal more damage than a support class, and has more consistent damage than a champions who nuke (usually AP champs). They'll become powerful if you feed them, but not as powerful as a carry.
If someone says they are playing a fighter as a carry, I would guess that they're basically wanting the team to feed him/her. It can be a viable strategy, but feeding a legitimate carry is usually an easier path to victory.
The main difference lay in the team's strategy. A fighter and a carry often have similar stats and mechanics for what the game knows but the team decide who's the carry and eventually a "fighter" or secondary carry. The carry is supposed to get more gold and exp than the others so that he can "carry" the team to victory, while the fighter just helps.
Often the fighter switch with the carry in the case that the enemy team has a very good counter for the main carry.
Carry can be defined into AP and AD. Usually they are glass cannons, going to an all DPS build. And of course, they are squishy.
Fighters are usually tanky DPS. They go for items that have health, defense, and damage, like Frozen Mallet. They are usually melee. Some examples are Xin Zhao, Renekton, Riven, and Nasus.