Map Awareness
Riot has garnered some pretty effective videos with their Video Tutorial contest (Links posted below). As a beginner, you want to primarily focus on making sure you don't get ambushed by the other team (called a "gank"). While some of this involves Map Awareness, you also need to learn to not overextend, as well as keeping an eye on the enemies you're fighting in your lane. If you cannot see one or both of the enemies in your lane, let your teammates know!
Last Hitting
Killing minions and enemy players nets you gold to buy items. However, if you don't land the killing blow on an enemy minion, you will get ZERO gold, even if you've done most of the damage to it.
The faster you kill enemy minions, the faster the minions reach the enemy tower, which will make quick work of all non-cannon minions. It's better to keep the minion fights as close to the middle as you can; you only need to finish off the minion to get gold from it -- see if you can limit yourself to only attacking minions when you will kill them!
Tower Aggro
The enemy towers are dangerous. They have lots of health, armor, and a powerful attack that can quickly decimate low-level heroes. It's important, then, to know how the turret picks its target.
Enemy towers will always target allied minions first, switching to champions only if there are no longer any allied minions around. The exception to this is if you deal damage to an enemy champion, in which case the tower will immediately switch to the damaging champion, shooting at them until they die or move out of range.
Let me repeat that: If you deal damage to an enemy champion within range of the tower, it will immediately start shooting at you: You do not want this. Be very careful when attacking the enemy at their own tower.
Recommended Viewing
Four of these videos are a result of the Valoran Video Contest held by Riot Games, and are quick, 1-minute videos touching on just the basics (which seems exactly what you're looking for). The fifth video is made by Shurelia, an employee of Riot Games, and goes much more in-depth into what "Zoning" is, and how to take advantage of it.
Movement prediction is a display option that only comes into play if you temporarily lose your connection to the game server. Without it, your champion will appear to stand still until you regain connection and receive your true position from the server. With it, your champion will appear to move to the location you last directed it to.
Best Answer
As of now, nobody has explained the differences between classes/roles. Classes have been explained fairly well elsewhere:
Tank: Very difficult to kill and usually low damage. Tanks normally have forms of innate ways to survive and/or crowd control abilities.
Fighter: Often good duelists. They do decent damage as well as are fairly tanky. Often used as splitpushers since in team fights they die quicker than tanks.
Assassin: Good at killing single targets very quickly and often able to get back out. Tend to be very squishy.
Mage: Generally have aoe abilities and might have a slight amount of cc mixed in their kit.
Artillery Mage: Special class of Mage that deals damage from long range. Normally have very low amounts of cc and/or less damage than their Mage counterparts.
Control Mage: Similar to Mages but have more crowd control mixed into their kit in place of damage. They can potentially contain high damage as well but not necessarily.
Marksman: (ADC) Long ranged physical auto attackers. Great at taking down towers or shredding tanks.
Now, Roles have not been explained well. In league of legends there are exactly 5 roles. Most classes can be a part of most roles but there are single exceptions.
Top: Solo laner who starts the game in the Top lane. Typically Tanks/Fighters are up here but pretty much any class can be played here.
Middle: Normally assassins, mages, or control mages belong here. This is a squishier lane that normally builds AP or AD but don't have great escape/survivability tools (besides assassins). The shorter lane allows for these types of roles since the towers are closer.
Jungle: Most classes can be played here but the most common are Tanks, Fighters, and Assassins. They stay in the jungle killing jungle camps for gold/exp and constantly roam into lanes to help their team get kills
Bottom (sometimes referred to as ADC): This is the most restricted Role. This is always played by a Marksman(ADC) because they have the protection of a support to keep alive.
Support: Supports are normally tanks or control mages. Supports have less of an income than any other role and so need to scale off of usefulness rather than items. Some tanks that are strictly good in the support role I might classify as a utility tank (Thresh/Blitzcrank: They are good at pulling people into their team and locking them down, but are a little less tanky than champions that are strict tanks.)
NOTE-There is some flexibility between classes. Part of class build often deals with how a champion is built. For example, Pantheon can be played both bruiser or Assassin. Full damage he is considered an assassin but if he builds some tankiness he becomes a bruiser.
EDIT: Apparently Riot has it's own group of additional classes. The classes I defined represent what the community Uses.