Who do you focus during a teamfight?

There are so many different factors to take into account here that it's really difficult to give a good answer here without more information. What hero/role are you playing, are you ahead or behind, what are the enemy heroes, etc?
Lets assume that you are a ganker with burst dmg (ie Nyx Assassin, Queen of Pain). You should evaluate the importance of the enemy heroes you are going up against. Is there a Luna w/o a Black King Bar that you can insta gib? Do it. She's the carry and can do some ridiculous dmg in a team fight. On the other hand do they have supports that have incredibly game changing abilities, such as Dazzle or Warlock? Removing them from the fight before Dazzle can shallow grave or Warlock can ult will mean that the fight will essentially be yours unless you mess up or they buy back.

The people you do not want to focus are the heroes that are happy to die once they get their spells off

An Earthshaker that has already blinked in and used all his spells could not care less that you turn around and figuratively blow your load on him, he's already done his job. Try to deal with the remaining heroes if you can still salvage the fight and deal with him once everyone else is dead.

Which heroes are the most important at which stage of the game/fight?

There is no easy way to teach or learn this in a general sense and mostly comes with greater game sense and hero/ability understanding. On a case by case basis it can be easier to define, so if you have a specific example I would be more than happy to go into greater detail for you.

The cooldowns of your enemies' skills

If there's a Tidehunter who hasn't ravaged, a Zeus who hasn't used  Thundergod's Wrath, etc. then you might want to focus them and kill them before they can get that ability off. If there's a Tidehunter that has  Ravaged, there's no point trying to kill him quickly because he's a low impact hero with tons of HP and damage reduction.

Mobility

When going into a fight with an advantage I like to focus on killing the enemies who have best mobility. Cutting off their team's ability to chase or flee can turn what would've been a minor win into a teamwipe.

Teamfight abilities

Some skills, especially auras, act as a magnifier for a team the longer a fight goes on. If your fights aren't fast, it could be worthwhile to focus the Vengeful Spirit, Luna, Necrophos, Elder Titan, Beastmaster etc., especially if they're carrying aura items or a Mekansm.