Every language lends itself particularly well to certain forms of rhythm. Greek poetry was often in hexameter, which was regarded as heroic. This meter was adapted by the Romans, but it doesn't always work for English. Throughout the ages, many poets and critics have proposed certain meters as being more "natural" (ie., better) for English than others.
The traditional rhythm, popularized by Shakespeare, is iambic pentameter. It is the most common meter of English poetry, at least until the rise of modernism. It is still the most instantly recognized meter, and is one of the easier meters to deal with, because it really does correspond to certain natural patterns in English speech. It's not unusual to find that you've just uttered a sentence in iambic pentameter purely by accident.
But, soft! What light through yonder window breaks?
Iambic pentameter is often touted as the "natural" meter of English. However, this meter doesn't always sound natural, even with the occasional foot-substitution (a trochee or dactyl for an iamb, for example). Therefore, some poets have sought natural-sounding alternatives to iambic pentameter.
The "cheap" alternative is free verse. If there's such a thing as "natural" meter to English speech, free verse captures it by definition; free verse means letting the line have whatever structure it has when you say what you have to say in the way it has to be said. It can be whatever you want it to be. Consider the first two lines of Allen Ginsberg's Howl, which is gripping and intense:
I saw the best minds of my generation destroyed by madness, starving hysterical naked,
dragging themselves through the negro streets at dawn looking for an angry fix....
One of the most powerful meters in the English language is trochaic tetrameter. It impacts on the instinctual level in a way that more subtle or complicated meters can't rival. This is the meter used by the poet William Blake in his most famous poem, The Tyger:
Tyger, Tyger, burning bright,
In the forests of the night;
What immortal hand or eye,
Could frame thy fearful symmetry?
In other works, Blake eschewed the distinction between stressed and unstressed syllables, and simply used unfooted tetradecameter, fourteen syllables to a line regardless of stress. This could be hectic and expressive or it could be adapted to a more conventional iambic heptameter for a more soothing effect. Consider the opening lines from The Book of Thel:
The daughters of Mne Seraphim led round their sunny flocks,
All but the youngest. She in paleness sought the secret air,
To fade away like morning beauty from her mortal day.
The Victorian poet Gerhard Manley Hopkins had a rather complicated meter, which he called "sprung rhythm." He claimed that it was the natural meter of the English language, but that it had had not been used in English poetry since the Elizabethan era. In sprung rhythm, the number of stressed syllables per line is fixed, but there could be any number of unstressed syllables inserted anywhere around the stresses. For example, in his poem God's Grandeur:
The world is charged with the grandeur of God.
It will flame out, like shining from shook foil;
It gathers to a greatness, like the ooze of oil
Crushed. Why do men then now not reck his rod?
Generations have trod, have trod, have trod.
And all is seared with trade; bleared, smeared with toil;
And wears man's smudge and shares man's smell: the soil
Is bare now, nor can foot feel, being shod.
In this stanza we can see a clear, if peculiar, meter: Four stresses through each of the first five lines, then five stresses for the sixth and seventh lines, and four again on the last line.
While there are many other meters, these are my favourites, because they really do capture the particulars of conversational English, both at its most flowing and its most forceful.