If there were no chains, Yu-Gi-Oh! would only have half as many rules questions.

Chains are created when a player activates a card or card effect. After this, the opponent may chain one of his cards, or, if he does not, the same player may chain a card to the first one.
This sounds a bit dry, but we'll handle this step by step.

Generally, it can always be chained to a card effect (unless it is permanent). But it is important to look for the Spell Speeds of the effects. Also, it is important to differ between the activation of a card or effect AND the resolve of a chain. You can only chain with an effect that has the appropriate Spell Speed.
Chain links can be added until there are no more cards that anyone of the players wants to add.
Besides reacting to the previous activated effect, you can always also react to the event that triggered the effect of the first chain link.

There is also a big difference in how the card itself is handled in comparison with the effect of a card. During the resolve of a chain, the effect is not dependant on the presence of a card on the field, in the graveyard... (the place, where the card was activated). For example, if the card was face-up on the field at activation and got sent to the graveyard before resolution, the effect resolves normally. The only exceptions to this are permanent Spell and Trap Cards that have no permanent effect, like "Ultimate Offering". These chain links resolve without effect if the card is no more face-up on the field during resolution.

Normally, a card is not sent to the graveyard at activation or while the chain resolves. Instead, all cards of the chain are sent to the graveyard at the same time after the whole chain resolved. The only exception is when another effect in the chain causes a card to leave the field due to the effect.

In addition, always remember that you cannot chain into a resolving chain!

1. The easiest example - one chain link

"Foolish Burial" is activated from your hand:

Chain Link 1: "Foolish Burial"

The chain only consists of the activated "Foolish Burial". Until now, almost nothing happened. The effect of "Foolish Burial" is not applied at this time, because both players may still react through chaining.
In our example, no one wants to chain, so the chain resolves. Now, a monster is sent from the deck to the graveyard. "Foolish Burial" is sent to the graveyard and the effect of it is the final result of the chain.

2. We add another chain link

Now we come to the typical question "What happens if "Mystical Space-Typhoon" is chained to card X ("Foolish Burial")?

The chain looks the following:

Chain Link 1: "Foolish Burial"
Chain Link 2: "Mystical Space Typhoon" -> targeting "Foolish Burial"

Because "Mystical Space Typhoon" targets, the target has to be choosen upon activation. Now, we resolve the chain. This always happens in reversed order of the creation of the chain:

Chain Link 2: "Mystical Space-Typhoon" destroys "Foolish Burial"
Chain Link 1: "Foolish Burial" sends a monster to the graveyard

Because the card text of "Mystical Space-Typhoon" does not include negating an effect, the effect of "Foolish Burial" resolves, even though the card itself is no more on the field.
After the whole chain resolves, all cards are sent to the graveyard. In our example, this is only "Mystical Space-Typhoon", as Foolish Burial was already sent to the graveyard by the effect of "Mystical Space-Typhoon".

3. A chain with 3 chain links

Player A activates "Limit Reverse", player B chains "Royal Decree" to which Player A chains "Dust Tornado", targeting "Royal Decree".

Chain Link 1: "Limit Reverse" -> targets a monster in the graveyard
Chain Link 2: "Royal Decree"
Chain Link 3: "Dust Tornado" -> targets "Royal Decree"

When resolving the chain, chain link 3 ("Dust Tornado") first destroys "Royal Decree". "Royal Decree" is sent to the graveyard. Because "Royal Decree" is a Continous Trap Card, it looses it's effect as soon as it is no more face-up on the field. This means chain link 1, "Limit Reverse", is not negated and the targeted monster is Special Summoned. After the chain fully resolved, "Dust Tornado" is sent to the graveyard - "Limit Reverse" is a continous card and remains on the field.

4. Not chaining into a resolving chain

Player A activates "Allure of Darkness" and chains "Dust Tornado" targeting Player B's "Call of the Haunted", which is in chain targeting "Hundred-Eyes Dragon".

Chain Link 1: "Allure of Darkness"
Chain Link 2: "Dust Tornado" -> targets set "Call of the Haunted"
Chain Link 3: "Call of the Haunted" -> targets "Hundred-Eyes Dragon"

Firstly, "Hundred-Eyes Dragon" is Special Summoned through "Call of the Haunted". Then, "Dust Tornado" destroys "Call of the Haunted". This causes the permanent effect of "Call of the Haunted" to destroy Hundred-Eyes Dragon. The reason for this is that continous effects do not have a Spell Speed and therefore do not use the chain. They can use their permanent effects in between a resolving chain.
Now it is time for the Trigger-Effect of "Hundred-Eyes Dragon". But because we are still in a resolving chain, the effect cannot be activated right now. Therefore, we continue resolving the next chain link. So player A draws 2 cards and banishes a Dark-Monster from his hand. Or, if not possible, sends his hand to the graveyard.
After this, the "Dust Tornado" and "Allure of Darkness" are sent to the graveyard.

So what happens to the non-optional Trigger-Effect of "Hundred-Eyes Dragon"? It is a mandatory effect. Because it was not possible during the resolve of the chain, he now starts a new chain automatically.

Chain Link 1: "Hundred-Eyes Dragon"

Both players may chain to this new chain. If no one wants to, the player gets to add an "Earthbound Immortal"-Monster from his deck to his hand due to "Hundred-Eyes Dragon's" Effect.

When several effects activate at the same time ("trigger"), we use SEGOC to determine the order of the effects. SEGOC is the only possibility that chains contain more than one effect with Spell Speed 1. They go on chain as followed:

• Mandatory effects of the Turn Player
• Mandatory effects of the non-Turn Player
• Optional effects of the Turn Player
• Optional effects of the non-Turn Player

If a player controls more than one effect of the above types, he may decide the order of his own effects himself.

Examples:

A player has 2 set monster cards and the opponent activates "Swords of Revealing Light".

• Both monsters are flipped face-up and their Flipp Effects trigger.
• The player controlling both monsters may decide the chain order.

Player A has a set "Ryko, Lightsworn Hunter" and a set "Morphing Jar". Player B activates "Swords of Revealing Light".

• Player A chooses which of the two effects goes first on the chain.
• If he decides "Ryko" to be chain link 1, then "Morphing Jar" is chain link 2.
• In the resolve, first both players discard their hand and draw 5 new cards due to "Morphing Jar's" effect. Afterwards, "Ryko" resolves as chain link 1.

Now, we look at a more complicated example that involves effects from both players.

Examples:

Player A attacks with "D.D. Warrior Lady" his opponent's "Ryko, Lightsworn Hunter". He decides to activate the effect of "D.D. Warrior Lady" after the battle.

• First, the mandatory effects of the turn player, meaning player A, form a chain. Because he has none, we go on to mandatory effects of the non-Turn Player.
• Therefore, the effect of "Ryko" becomes chain link 1 and the player targets the card he wants to destroy.
• Afterwards, the optional effects of the turn player are added to the chain, so "D.D. Warrior Lady" becomes chain link 2. The non-turn player does not control any optional effects.
• Resolve: "D.D. Warrior Lady" removes both monsters.
• If the targeted card was "D.D. Warrior Lady", the part of destroying resolves without effect. If the target was another card, the targeted card gets destroyed. Either way, the 3 cards are sent from the deck to the graveyard.

Player A has 4 parts of "Exodia" in his hand and attacks a "Sangan" with his own "Sangan". Player B also has 4 parts of "Eoxdia" in his hand.

• Chain Link 1 is player A's "Sangan", as he is the Turn Player.
• Chain Link 2 is Player B's "Sangan", because he is the non-Turn Player.
• Because a chain resolves backwards, player B's Sangan resolves first. He adds the 5th part of Exodia to his hand and wins the game. As he met a win condition, the chain does not further resolve.

When following this rule, there should be no problem of building the right chain for simultaneuos effects. There are also lots of effects that seem simultaneus, but are not.

Examples:

Player A has a "Treeborn Frog" and a "Sacred Phoenix of Nephthys" in his graveyard. It is his Standby Phase. Player B has a set "Torrential Tribute".

• The 2 effects DO NOT form a chain after the principle of simultaneous effects, because they do not activate at the same time. Both activate -sometime- in the Standby Phase.
• Player A may decide which of the 2 cards he wants to Special Summon first.
• Player A decides for his "Treeborn Frog".
• In a new chain, "Sacred Phoenix of Nephthys" is Summoned and activates its effect to destroy all Spell and Trap Cards on the field. Player B chains "Torrential Tribute" to destroy all monsters on the field.
• Because we are still in the Standby Phase, Player A may decide to Special Summon his "Treeborn Frog" again.

You will find more about this topic in "Priority in Phases".