If you need to enter the end, see my guide on how to get there.
Now that you are in the end, you will see that you have spawned on an obsidian platform. This can be either floating above the void or completely underground. DO NOT place any items on this platform, especially chests, as they will despawn if you return to the end later. Make your way to the island surface and you will see multiple obsidian towers that the dragon flies around. On top of every tower is an end crystal that you have to destroy. These crystals heal the dragon, so it is impossible to kill it without destroying these.
If you have a bow and arrow, you can try to shoot these crystals to destroy them from afar. Otherwise, you will have to build your way up to the top of each tower and hit the crystal to destroy it. (Two of the taller towers have iron cages surrounding the crystals, so you have to build up anyway). If you need more blocks, you are surrounded by useful end stone blocks. Be carful when climbimng the towers as the dragon can sometimes knock you off!
After destroying the crystals, it is time to fight the dragon. A lot of people think you need a bow and arrow for this, but you do not. The dragon likes to go to the middle bedrock structure and sit for awhile, giving you a chance to whack it with your sword. If you build a two block high pedestal underneath it, its bad breath will not be able to hit you here and it is easier to hit it. Now just do this until it dies!
Note: If you are also planning on respawning the dragon, do not tamper with the obsidian towers themselves. You have to build a tower next to the obsidian.