This explanation may be biased in favor of my musical background. It probably wouldn't get an A+ on a creative writing tests for an exact definition of meter, but it'll get the point across.
The meter of a poem is the rhythm in which the words flow. If they create a consistent rhythm, then the poem feels coherent and well put-together. In a metered poem, the lines are like measures, with the phrases being like notes -- they fit together in a certain pattern to create that whole.
For example, In a poem like a Shakespearean sonnet, a phrase is an unaccented syllable and then an accented one (da-DA, da-DA, da-DA), and that means they're iambic; pentameter means there are five phrases in a line ("Shall I comPARE thee TO a SUMmer's DAY?/Thou ART more LOVEly AND more TEMperATE/Rough WINDS do SHAKE the DARling BUDS of MAY/and SUMmer's LEASE hath ALL too SHORT a DATE ...")
This poem (one of my all-time favorites) is really useful for learning about meter. Each line is written in the meter it describes.
Trochee trips from long to short
From long to long in solemn sort
Slow spondee stalks, strong foot, yet ill able
Ever to come up with dactyl trisyllable
Iambics march from short to long
With a leap and a bound the swift anapests throng.Pronounciation Note:
Trochee - "TRO-shay"
Spondee - "SPAWN-day"Just read that aloud a few times, paying attention to how the words are accented, and maybe exaggerating them a little, and you'll start really feeling the rhythm of the different sorts of meter.
For the record, the hard ones to tell apart are anapests/trisyllable, and iambics/trochee. The difference is whether the poem's lines start with an accented syllable (trochee, trisyllable) or the unaccented ones leading up to it (iambic, anapests).
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Now, apply this to a poem like ... ummm ... "The Night Before Christmas."
"'Twas the night before Christmas, and all through the house,
Not a creature was stirring, not even a mouse."
Rewrite/say it as:
"Twas the NIGHT before CHRISTmas and ALL through the HOUSE
Not a CREAture was STIRring, not EVen a MOUSE."
See how that's similar to the line "With a LEAP and a BOUND the swift ANapests THRONG"? Two unaccented, one accented, da-da-DA da-da-DA da-da-DA ... Just like a 6/8 measure of 8th notes with a two-eighth-note pickup.

But I digress.
Then, you divide the line into its phrases, broken up based on how the meter works, like this:
"Twas the NIGHT/before CHRIST/mas and ALL/through the HOUSE,
Not a CREA/ture was STIR/ring not EV/en a MOUSE."
Four phrases per line, so that makes it quadrameter ... and ta-da! Anapestic quadrameter!
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See? It's easy.
