I've been learning a great deal about meter in the past few weeks, and therefore this little bit of information excites me. An anapest is a tri-syllabic metrical "foot", a section of meter with three syllables, and tetrameter simply denotes four-footed meter, which makes it then four three-syllable sections on each line. English verse relies on syllable stresses to define the metrical feet. Thus, an anapest in English is two unstressed syllables followed by a stressed syllable: "dum dum DUM", if you will.
Anapestic Tetrameter is much like a series of Seussian four-footed creatures -- long gangly legs, fat slanted eyes, and shaggy blue fur. Each foot has three toes, the last toe being a big toe, perhaps with a claw. And the creature is loping along a Seussian landscape, rolling hills and tall tufted trees, at a gallop.
Here are two lines, for illustration, from "If I ran the Zoo" (notice how the first foot only has two syllables; dropping a syllable here or there is a common practice in any sort of meter):
"the world's biggest bird from the island of Gwark,
who eats only pine trees, and spits out the bark."
Imagination, meet Perfect Rhythm. Copulate. Make children.
Classically, this form wasn't used much for, you know, serious poetry, since it does roll along rather comedically and joyfully. Lord Byron, however, gave it a go and wrote "The Destruction of Sennacherib" in anapestic tetrameter. Honestly, messieur Byron, you're good, but you're not the Seuss.
The Destruction of Sennacherib
The Assyrian came down like the wolf on the fold,
And his cohorts were gleaming in purple and gold;
And the sheen of their spears was like stars on the sea,
When the blue wave rolls nightly on deep Galilee.
Like the leaves of the forest when Summer is green,
That host with their banners at sunset were seen:
Like the leaves of the forest when Autumn hath blown,
That host on the morrow lay withered and strown.
For the Angel of Death spread his wings on the blast,
And breathed in the face of the foe as he passed;
And the eyes of the sleepers waxed deadly and chill,
And their hearts but once heaved, and for ever grew still!
And there lay the steed with his nostril all wide,
But through it there rolled not the breath of his pride;
And the foam of his gasping lay white on the turf,
And cold as the spray of the rock-beating surf.
And there lay the rider distorted and pale,
With the dew on his brow, and the rust on his mail:
And the tents were all silent, the banners alone,
The lances unlifted, the trumpet unblown.
And the widows of Ashur are loud in their wail,
And the idols are broke in the temple of Baal;
And the might of the Gentile, unsmote by the sword,
Hath melted like snow in the glance of the Lord!