There is a type of puzzle called an acrostic which is a completely different beast. You answer clues. Each answer letter has a number (not in order). You put the letters in the order of the numbers and it spells out a quote or a joke.

The OP's thing is also called an acrostic. For example:
An Acrostic by Edgar Allan Poe
Elizabeth it is in vain you say
"Love not" — thou sayest it in so sweet a way:
In vain those words from thee or L.E.L.
Zantippe's talents had enforced so well:
Ah! if that language from thy heart arise,
Breath it less gently forth — and veil thine eyes.
Endymion, recollect, when Luna tried
To cure his love — was cured of all beside —
His follie — pride — and passion — for he died.

An acronym is a word made from the first initials of the phrase it stands for - sort of the opposite process. It is not a word in itself beforehand, e.g. Self Contained Underwater Breathing Apparatus = scuba.