What is the meaning of the word poof?
The word "poof" is used to indicate something vanishing, but in the following usage of the word, nothing has disappeared. Is this its proper usage and if it is, could you please explain it for me? Also, is it possible to restate it in different term?
"I roughly cut around the pictures and placed them between two clear adhesive sheets, and laminated them! I then cut them all out, and poof! My new business cards are done." (colormekatie.blogspot)

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Poof in this context means "like magic" it appeared.  It comes from the concept that a magician will say the magical word poof and something will disappear.  However, it can be used for "magical acts" of appearance or disappearance.
To restate the sentence: I cut them all out and like magic, my new business cards are done.  
You could also use two other words instead of poof to indicate a magical event.  "Presto" or "Abracadabra".

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"Poof" is an interjection as used here, to express a thing happening within seconds. A very dexterous person does something very skillfully, and to others that seems like a magic. 'Poof" shows that level of magic like skill. This is my opinion.

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I found another sentence: The fairy waved her wand and poof - the wife had tickets in her hand for a world cruise! 
and this is definition: 
poof   noun 
poofs, plural; pooves, plural; poufs, plural

1.An effeminate or homosexual man

exclamation /po͞of/  /po͝of/

1.Used to convey the suddenness with which someone or something disappears
- once you've used it, poof—it's gone

2.Used to express contemptuous dismissal
- “Oh, poof!” said Will. “You say that every year.”

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Your understanding of the word is right, so it's hard for me to see the example you mention as correct. Maybe the writer means that the material he cut out disappeared.