Late game is the most impotent phase of the game. What is important in the late game phase, and what are the most valuable objectives?
Late game starts around 25:00 and ends when you or your opponent destroys the "nexus", the "nexus" is in the heart of your base and the enemy's can't kill it unless they have at least one of your "inhibitors" down and destroyed the two towers protecting the "nexus", but the same applies to you. As you level up the time it takes for you to re-spawn when you die gets longer, this is why team fights at this point are game changers. The time it takes to respawn after dying in the late game phase is enough time for the winning team to take objectives that will help them later on.
The most important thing in the late game phase is to be winning the team fights, even if you don't kill their whole team if your team is able to kill more of them then they have killed of your teammates that's winning the team fight. Normally if you are able to catch and kill one opponent then you can try to get objectives a lot easier knowing that it is a 5v4 for a small amount of time. Being the first team to destroyed an "inhibitor" is the key to getting other objectives easier because once you do so it will spawn 1 super minion per wave in the "inhibitor" lane that you destroyed it in.
How to win a team fight, depending on your "role" is to kill their carries or protect/"peel" your carry's or "engage" the team fight with the odds in your favour. Taking out the enemy's carry's first is a great start to winning the team fight, but you also want to keep your carry's alive because they will be the highest damage output on your team. If you are the carry you will want to stay alive and deal the most possible damage to the opponent team, to stay alive you will want to deal damage to whoever is hitting you and stick with your team so they can help you kill them.
The objectives that are most valuable in helping you win the game would have to be, "inhibitor", "towers", "Baron" and "dragon". Although you do not need to kill "Baron" or "dragon" to win the game it gives your team a buff that can help win team fights and in turn may help win the game. "inhibitors" do re-spawn so make the most out of it then you get it down, The baron buff last 3:00 minutes or till you die. The "elemental dragons" stack up but at 35 minutes the "elder dragon" takes the place of the "elemental dragon".The "elder dragon"buff last 1:30 minutes or till you die.
Note: That "Baron" or "Dragon" can be stolen by the opponent team by last hitting it, only do these objectives when you know there is very little chance of them doing so. Having a jungler with "smite" will help you minimize the chances of them stealing it, if the jungler uses "smite" to last hit the "Baron/Dragon"."smite" is a summoner spell that deal true damage to monsters or minions.