When a spell, trap, or monster effect is activated, this starts something called a chain. Summoning a monster can also start a chain. During that chain, each spell, trap, or monster effect used each have a spell speed that determines if they can actually be part of that chain. Here's the breakdown for spells/traps:
Regular spells(Monster reborn, Dark hole, etc.) are spell speed 1. So are field spells, continuous spells, and equip spells. They can be chained to other spell speed 1 cards only, and can only be used in main phase.
Traps and quick-play spells(lightning bolt in upper-right corner, such as mystical space typhoon or Forbidden Chalice) are spell speed 2. They can be chained to spell speed 1 or spell speed 2 cards. Also, they can be used in any phase, and from your hand if it is your turn. During your opponents turn, they can be used if they are set.
Finally, spell speed 3 cards are counter-trap cards, such as dark bribe or solemn warning. They can chain to all other spells/traps, and can only be chained to by other counter-traps. Since they are traps, they have to be set before being used.