Skeletron

Information
Skeletron is a boss which can be 'summoned' by talking to the Old Man in front of a Dungeon at night. The Old Man will turn into Skeletron immediately and start attacking you, so be prepared when this happens. This is a difficult boss, do not try to fight him unless you have lots of potions, end-game equipment and throwable objects. Rapid attacking weapons work best, for example the Minishark or the Flamarang. A flail like Ball 'O Hurt (or even better, the Sunfury) will also work if you manage to hit him.

When you defeat Skeletron, you will gain access to the Dungeon. If you try to go into the dungeon by going through the door or digging yourself in without defeating Skeletron first, he will constantly spawn heads that will chase you throughout the dungeon. Note: these heads are full sized skeletron heads that will spin as their only attack, the longer you stay in the Dungeon, the more that are spawned. During the day these will insta-kill you, at night however they will do the standard damage as if you were fighting him as a boss. Note: If day breaks while you are fighting him normally, he will revert into a skull and insta-kill anyone near him.

Hands
He will mainly sweep his hands across the ground, or swing at an angle at a specific location (indicated by him raising his arms off screen diagonally). he will do this whenever you are not in melee range of his head. This attack deals less damage than when attacking with his head, but he will have more range to attack you that way.

The hands have 1500 Hitpoints each.

The Cloud in a Bottle accesory makes his sweep attacks easier to dodge

Head
Skeletron's head will begin to spin rapidly and home in on a specific player, attempting to stay on that player for as long as possible. Using a combination of grappling hooks and jumps, the player can outrun, or avoid, it as soon as it begins, but it is very difficult to get out of once Skeletron is on you. This attack is signaled with a large roar (similar to when a boss spawns) so you'll know it's coming.

Tactics
There are a few things to mind when about to challenge this boss.

First of all, you should make sure that you have a big, even surface to face him on. Building a long line of dirt above and beyond the dungeon's entrance is a good idea. Just make sure that you can get there fast with Skeletron on your tail.

Secondly (if you have the firepower) you could take out his hands first, as that makes killing the head so much easier. After the hands are dead and the head starts rolling towards you, it's easy to just run slightly ahead of him and hitting him with a long blade (like Blade of Grass or Fiery Greatsword).

Another great tactic to use is a Phoenix Blaster with about 500 Meteor Shots. Go to the top of the dungeon entrance and spam left click while aiming for his head.

Lastly, bring alot of potions and keep the pressure up. You only have until dawn before he gets bored and kills you and everyone else involved.

Respawn
Skeletron is currently the only boss that can only be summoned once per world. If you kill him you may enter the dungeon without the danger of skulls being summoned.

If you fail or Skeletron accidently goes too far off screen then you'll have to wait until another day to talk to the Old Man. There have been people who report that the Old Man does NOT reappear if you fail to kill Skeletron. Skulls still attack inside the dungeon however. It is also possible to kill the Old Man with a meteor. It's also possible for the Skeletron to respawn at night. Just be careful of the Meteorheads that spawn around where the meteor hits as these could kill you while facing Skeletron.

Also, a good strategy if you play on a multiplayer server is to build a miniature house with a Bed inside it. That way, you can respawn into the fight when you die. Just remember to set your spawn to that bed. It is possible for Skeletron to spawn past the dungeon. When this happens, only his head will appear. The case this is most in is in corruption caverns. This may be a possible bug

Trivia:

In the Alpha version, the monster had 3000 health instead of 6000