First the run down for those of you that do not know what a skillchain (also known as Renkei) is.

First and foremost, a skill chain is carefully planned and timed use of your weapon skills to cause special add-in effects to occur. Why is this important you may wonder? Well, very well laid out chains can massively boost your damage and help you take enemies that you normally would stray from at your current level.

How they work
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Simply put, you wait for 3 seconds after you see the person before you in a chain ready their weapon skill. Example
[player1] readies Fast Blade.
*wait 3 seconds*
[you] ready Burning Blade.
Skillchain effect: Liquefaction

Now, granted, it's kind of annoying to manually wait 3 seconds for you to give the command, so cut out the middle time and make the game do it for you. Ready a macro like so, and you're party will know what's coming and have time to prepare if you're leading off, and if you're the follow up, you simply hit the macro as soon as you see "[player1] readies [weapon skill]"
/p [ws name] in 3 seconds.
/wait 3
/ws "[ws]"
/p
The call in there being for an auditoury note so people that may not of seen the message in battle spam still know to hit thier respective macros now.

The chains themselves
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Okay, to be completely honest I'm not going to even attempt listing them all, as there are a metric ton. If someone in your party has a printed cheat sheet or the strategy guide, have them look up what skills you can combo out of your current usage. Basic ones, and the most commonly used are like:

Fast Blade -> Burning Blade
Red Lotus Blade -> Fast Blade
Flat Blade -> Red Lotus Blade -> Fast Blade

Magic Bursts
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A magic burst is additional damage from a spell that is cast 3 seconds after a skillchain effect (like Distortion or Liquefaction) has occured. Note that the START of the cast needs to be 3 seconds after, not necessarily the spells impacting. (Unless cast time exceeds 2 seconds, in which case you shouldn't be trying to chain it anyway.) The spell needs to be of the same element as the skillchain effect was, so have your coordinater check up on suggested magic for the BLM to use.

Tips and Suggestions
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Make sure to have a coordinater, and if you have a full party, it's sometimes a good idea to make pairings of how it will go, depending on composition. Another pointer, if you have any thieves past 15, try and make sure they end the chain with sneak on. (Special macro needed, will cover shortly) A good example would be a 3 man chain, 2 sword using wars and a thief.
Flat Blade -> Red Lotus Blade -> Sneaked Fast Blade

The reason for having a thief sneak and WS is that not only will their damage of the WS (granting it can stack with Sneak attack, they'll know in advance most times) is that the damage of the skillchain effect it creates is greatly enhanced too. Here is an example of the difference:
Fast Blade (40 dmg) -> Red Lotus Blade (50 dmg) -> Liquefaction (10 dmg) -> Fast Blade (40 dmg) -> Liquefaction (8 dmg) = Total dmg 148

Now, in that combo you did get an additional 18 dmg going, which isn't bad. But you can do much better. Same chain, but with sneak on the end.
Fast Blade (40 dmg) -> Red Lotus Blade (50 dmg) -> Liquefaction (10 dmg) -> Sneaked Fast Blade (130 dmg) -> Liquefaction (38 dmg) = Total dmg 268

You're getting the point by now. This chain get nastier as well as you level further. Anyone that has seen a lvl 30 thf skillchain with sneak+trick has seen some nasty damage laid onto the poor mob they're killing. To give you an idea, in Qufim on Acro's, I got a Trick+Sneak+Fastblade (a Yokodama) on the Tail end of a Raiden Thrust. Raiden did 70 dmg, my yokodama did 341 dmg with a 178 dmg Distortion effect. (Incase you're wondering, it instagibbed the poor acro)