Generally, each team has one jungler, which means that there will be one lane of two, and two lanes of one. Since the middle lane will always be a solo lane, this means that top and bottom will have varying numbers of champions, depending on the lane.
Top is usually considered a solo lane because the bottom lane is closer to Dragon, an important mid-game objective. Dragon is hard to solo for most non-junglers, and by keeping 2 champions in the bottom lane, it becomes quite viable to 3-man Dragon after a successful gank (or simply a double-kill). While top lane is closer to Baron Nashor, it takes quite a bit more effort to kill him - more than 3 champions in the early game can probably afford to put out. By the time Baron becomes available, the game is usually nearing the end of the laning phase, after which the difference between top and bottom lane is moot (because all 5 champions are roaming).
With a top solo lane, the champion has a leveling advantage and the potential to farm more minions (simply because there is no allied champion to compete with). Additionally, one must consider that if the other team does not have a jungler, they need to be able to successfully last 2 vs. 1. Most often, this means champions with good sustainability are better suited to the lane, such as Cho'goth, Mordekaiser, Vladimir, or Nasus.
Conversely, there are some champions who do not do well in a solo lane - these tend to be support champions like Soraka, Sona, or Janna who perform much better with a partner to buff/shield/heal. These champions usually go to the bottom lane to ensure that they are in a 2 vs. 2 situation.
In instances where there is no jungler on either team, both top and bottom will have an even number of champions. In this case you may be asked to "switch lanes" not because one lane is "better suited" than the other, but because switching lanes means changing which enemy champions you're fighting (and presumably, have been dying to).
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.
Best Answer
The term for this is Stutter stepping.
This is a hugely important aspect of many games for several reasons.
The reason it is important is because moving doesn't affect your auto attack cooldown. So if you can attack every second, in between those seconds, you can move and it won't delay your attack. This mechanic allows you to move in between your attacks without lowering your DPS (if done perfectly, it's normal to reduce it slightly).
Now when you're attacking a melee champion for instance who is coming towards you, you can attack them, run away in this auto attack cooldown and then attack again. This allows you to get in more attacks before they reach you.
Similarly you can stutter step towards an enemy that is retreating in order to get more attacks in.
Another reason is that it allows you to reposition yourself. In a team fight position is hugely important, you don't wanna group too much with your allies or be too far away from them. So in order to not lower your dps, stutter step.
The final reason that I can think of off the top of my head is that it allows you to dodge attacks/ be more unpredictable. If you just stand there attacking the enemy will hit you with their skillshots everytime. You can reduce the chance of them hitting you by stutter stepping in any direction.
It's important to note that you can start to move before your attack animation has finished.