A short glossary of terms first:
AD - Attack Damage (One of the two damage types. Armor reduces damage from AD.)
AP - Ability Power (One of the two damage types. Magic Resist reduces damage from AP.)
CC - Crowd Control (Any effect that interferes with the enemy's ability to control their character)
ADC (officially called "Marksman" by Riot, but the community prefers the term ADC, Attack Damage Carry)
The ADC's role is to provide high, sustained damage over the duration of a team fight. Usually from long range. They generally have low burst and terrible defense so they rely on their team to protect them while enabling them to get into position to deal their damage. ADCs are almost exclusively seen in dual lanes with a support, much more rarely in the jungle (usually only when the lanes provide a lot of CC). See Ashe, Sivir, Lucian, Jhin.
Assassins specialise in dealing a high burst of damage to a single target, and usually have tools that enable them to escape afterwards.They don't have good sustained damage, and usually they're very close range. AP assassins are more common, but there are good examples of AD assassins. In team fights, Assassins aim to eliminate a high priority target (ADC or Mage) as the fight starts. Depending on the champion, they might be played mid, jungle or top. See Leblanc, Zed, Ahri.
Tanks specialise in survivability and screening for their team. The tank either prevents the enemy team from attacking his carries with CC, or forces them to attack him by diving deep into the enemy team and threatening their carries. Tanks have excellent defensive stats, usually a powerful suite of crowd control, and decent mobility. Most (but not all) tanks have low damage and range. Tanks are usually played in top, jungle or support. See Trundle, Tahm Kench, Gnar, Maokai.
Mage / Mid (I have very rarely seen "APC" used to refer to Mages but I guess it happens)
Mages deal moderate sustained damage over the duration of a team fight. They usually have a lot of area effect spells that allow them to spread damage across a whole enemy team, and some good CC. Like ADCs, they tend to be squishy. Although most Mages deal AP damage there are some notable "AD Casters" whose skills scale with AD rather than AP. Mages are almost always played Mid (to the point that the role is often called "Mid") but also show up Top. See Viktor, Varrus (AD Caster), Aurelion Sol, Vladimir.
Utility Support (usually called "Support")
Supports aid their team either sustain or crowd control. They often resemble mages in terms of defence & mobility but with far lower damage. Supports might specialise in engage (CC or movement abilities that allow their team to force a fight), disengage (the opposite, always letting you run away), or sustain (healing). There's a lot of variety within this group so it's hard to describe their role in a teamfight clearly, but usually it's disabling or enabling key targets. While Supports are usually run as part of a dual lane, a couple make it into the mid (usually before being nerfed back to bot lane). See Thresh, Bard, Karma, Lulu.
Assassin Tank Mage Carry
Usually called "Lee Sin"
These terms would, I think, describe most champions you're likely to run into, and they're all terms that I hear used regularly in the community. I've never seen wide uptake of stuff like "wardens" or "vanguards".