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.
Alright, I pulled these up with a LoL Replay of a Dominion game.
Bonuses Granted:
The character model grows by roughly x2
The first grants an absorption shield with 212.5 (+12.5 x level) health that will recharge if the champion doesn't receive damage after 10 seconds.
The second buff allows the champion to passively detonate a lightning blast inflicting 90 (10 x level) magic damage on the champion's autoattacks and damaging spells with a 4 second cooldown.
Spawn Time: 3:00 Minutes
Channel Time: 5 seconds
Respawn Time: 3:00 Minutes
Duration: ~45-60 Seconds (Not positive on this one, hard to find a game where someone kept it the whole time.)
Source: League of Legends Wikia
Best Answer
So let's get the obvious thing out of the way:
Yes. It is indeed very possible to destroy the two nexuses at the same "time."
That's because of how "time" works in computer games: discrete intervals that are by no means "tiniest amounts of a second".
This is how most game servers are written, due to the nature of modern computer networks being packet based (that is, in bursts of bits as opposed to a continuous flow of data):
This loop can be implemented in many ways — especially the "update the world accordingly" part — but the fact remains that this is a loop, and it only runs so many iterations every second.
A typical performance targets here is 30 or 60 updates (or ticks) per second, or an update within 33 or 16 milliseconds (that's short, but certainly not microseconds short!). So long as the Nexuses... Nexi... final bases go down in the same tick, then they've gone down at the same "time." Example with made up numbers:
What Nexus died first here? You can't tell.
Further complicating the matters here is network lag and compensation thereof. From the Valve Developer Community comes this handy graph for the Source Engine:
Since packets take time to reach the clients, this means that the clients are playing in the past (and the amount of time they do come in the past depends on their ping), and their user inputs come from even further back. As a result, the server might have to essentially "rewrite history" around them.
In one player's "user input tick 105" packet, you can claim to fire a bullet directly through another player's skull. This other player's "user input tick 105" packet arrives a few moment later, claiming he stepped to the right. Whoops!
Of course the server can't keep things in limbo for too long; there is a grace time after which the server must finally update the world state once and for all, and if the second player's packet arrives too late, then the shot will still count as a headshot.
Nitpickers' corner. Actually in this case, the Source Engine will award the headshot no matter what; this is why heavies for example can kill you "through" walls.
So, at the end of the day, it still is possible to have both nexuses go down at the same time.
Now, since the Nexus can't move around, there is a bunch of optimizations that can or can't be done that muddle the waters significantly more. Unfortunately, this logic is all on the server-side of Riot's; unless a group of 10 players sufficiently close to their data centers can dedicate significant amount of time to SCIENCE, there is little hope that we can figure out how League of Legends actually resolves ties, as it's rather blindingly obvious that the game does not allow ties.
No. It is blindingly obvious that you cannot draw a game.
Riot boasted that in January 2014 27 million players played League of Legends daily. Thus, the lower bound for the number of games played on the service daily is 2.7 million.
Nitpickers' corner. Depending on how you define "player" it is perhaps possible that some of the 27 million players won't actually play at all and just log in to check their profile page or the tribunal or whatever it is that LoL players do. On the other hand many more will play more than one game. At any rate, I don't have a better number to work on.
Now, it is true that I cannot find any report about the possibility of drawing League of Legends games on the internet. This discussion has happened already on the League of Legends forums multiple times (EU NE forums, EU W forums, NA forums, ...); if some player did experience a draw, we would know about it.
Let's guesstimate the odds:
That means the number of draws that would happen every day is 2.7 million × 0.01 × 0.00001 ~= 3. There would be 3 draws per day. The game as grown in popularity over time since 2009, so let's say that averages out to just 2 draw games per day. The game has been out for 1,768 days today, so that's 3,536 draws since ever.
How come we know of no single draw, then?
If draws are so rare, they would make for an excellent YouTube video at least, or for some interesting screenshots. "You won't BELIEVE how this League of Legends game ended!" So you can bet that people would post about them, write about them, upload videos of them. If nothing else, people would write about it in the threads that have come up earlier. Additionally, videos like this would include lines about the game ending in a draw, together with all of the unusued lines, but they don't.
The silence is deafening.
So what?
The only rational explanation is that there can't be draws in League of Legends and that the game server, somehow, resolves ties and awards victory to one and only one team.
There are several ways in which this could happen, and we are unlikely to know which one is being used, everything else being equal:
If I had to guess, I'd agree with a variation on the team "favour team X over team Y every time", as it's the simplest solution to deal with a situation you clearly don't want or can't handle.