It is split into 3 lanes top,mid and bottom then there is the support and jungle. The support goes to the bot lane with another player while the jungle is taking the camps inbetween the 3 lanes.
The goal is to take out the turrets, then nexus (crystal looking thing) of the other team. However there are side objectives in the game such as dragon which spawns after 10 minutes of the game starting and 5 minutes after each time it dies.
The dragon gives buffs depending on which dragon it is such as the infernal dragon which gives +4% bonus damage these effects do stack up to 4 times. But the first 3 dragons are random the rest of them are the same as the third dragon and if you get 4 dragons ou get the soul of whatever that 3rd dragon was which gives you an extra boost. After the soul is claimed however the elder dragon will spawn, which executes low hp enemies as a reward of getting it, this effect only last a minute or two, and like usual after 5 minutes another elder dragon will spawn.
The main way to get stronger is through levels and items you get both from killing little ai minions (camps if jungle) that if you get the killing blow on give you gold aswell as xp, but if you are only around them when they die you get only xp.
gold is used to purchase items which depending on who you are changes what itmes you will purchase. There is attack damage (AD) and ability power (AP) characters will build one or the other depneding on who they are (a handful can build both) or of course the tanks that build magic resist (mr), armor or health. Magic resist is good against ap, armor against ad and health is well health.
It would be broken if they didnt have a counter so luckily they do. Armors counter is armor pen, magic resists' counter is magic pen and healths counter is true damage which goes through all of the above.