Continuous Trap Cards are Trap Cards that stay on the field after activation. The effect(s) of these cards will remain active until the owner either cannot maintain a cost or condition, or it is destroyed. One example of a Continuous Trap Card is "Spellbinding Circle", which effectively freezes an enemy monster on the field, so it can neither change battle position, nor declare an attack.
Most Continuous Trap Cards require a cost to maintain, or possess a conditon under which the card is destroyed. Two Examples of these are "Imperial Order", and "Call of the Haunted". "Imperial Order" requires a cost of 700 Life Points to maintain, but "Call of the Haunted" is destroyed when the monster used with it is destroyed.
Another such example of a Continuous Trap Card is "Royal Decree", which negate's Trap Card's effects. However, since it's still Spell Speed 2, it cannot be used against cards like "Magic Drain", "Seven Tools of the Bandit", "Negate Attack", or "Trap Jammer", since they are all Counter Trap Cards, and thus all Spell Speed 3.
You can activate Continuous Trap Cards, and activate their Ignition-like effects on the same Chain. For example, when your opponent activates "Monster Reborn" you can activate "Royal Oppression" and activate its effect on the same chain.
Also, cards such as "Chain Burst" and "Curse of Darkness", have to be face-up on the field before the card it inflicts damage against is activated; you cannot chain these cards to the activation of the card type mentioned, and still maintain the effect of inflicting damage. This is because the timing has been missed; the card in question has already been activated.
Some continuous trap cards also become monster cards when activated; like Stronghold the Moving Fortress, which gives you more defense when you have no monsters.