The Basics: Scansion, Meter, and Rhyme

Syllabics v. Scansion
When we scan lines of poetry (or even prose), we may begin by first looking for syllables. For example, here's a line from Robert Frost's "Stopping by Woods on a Snowy Evening":
Whose woods these are I think I know
Here's how this line looks with syllables (with X representing a syllable):
Whose woods these are I think I know
X         X        X       X  X X    X X
But when we scan, we are looking for rhythm, for the stressed (/) and unstressed (-) syllables. So here is that line as a scanned unit:
Whose woods these are I think I know
      -     /             -    /    -   /      -   /    
There's no perfect way to get a handle on scansion, you just have to keep trying to hear the rhythm of words and phrases - always being aware that local dialects and diction change the way that we hear words.

Meter
There are, however, names for certain types of rhythms that seem to occur over and over again. We call these "feet" and, Paul Fusell, in Poetic Meter & Form, defines a foot as "a measurable, patterned, conventional unit of poetic rhythm" (19). Here's a quick run down of the most common feet (and their fancy names):
iamb ("iambic"): - / (below, above) (probably most common in English)
anapest ("anapestic"): - - / (understand, interfere)
trochee ("trochaic"): / - (ugly, pretty)
dactyl ("dactylic"): / - - (merrily, tenderly, beautiful)
spondee ("spondaic"): / / (hum-drum)
pyrrhic: - - (of the, in the)
As you might imagine, it often takes many feet to compose a line of poetry. The number of feet in a line will determine the ultimate rhythm of that line. These numbered feet also have fancy names:
monometer - one foot
dimeter - two feet
trimeter - three feet
tetrameter - four feet
pentameter - five feet (and probably the most common in in English language verse)
hexameter - six feet
hectameter - seven feet
octameter - eight feet
Here's a line from Ralph Waldo Emerson's "The Problem":
These temples grew as grows the grass
   -       /     -     /      -     /        -     /
The foot pattern is iambic and there are four feet, so this line is iambic tetrameter

Rhyme
Many beginning poets think that there is only one way to rhyme, the "true rhyme" we find in song lyrics, limericks, and nursery rhymes. However, there are many more subtle ways to use rhyme to add musicality and fluidity to your poems. Here are a few of the most common rhyming patterns you may find in poems:
True Rhyme - identical sound patterns (i.e. cloud/loud, tree/free)
False Rhyme - identical consonant sounds (i.e. insist, grace, blessed)
Slant Rhyme - similar sound patterns (i.e. God, dodge, ugh)
There are other sound patterns that can also be used to produce rhymes:
Assonance - repeated accented vowel sounds (i.e. speed, greed, weed, lead)
Consonance - repeated consonantal sounds (i.e. lack, back, brick)
Alliteration - repeated initial stressed sound (i.e. super, sonic, submarine, sailing)
Note: Rhyme and Meter can work together, but they do not have to. Sometimes you can create a rhyme without worrying about the meter (this really depends on the types of devices you are using to make the rhyme). A line can have a "rhythm" without having a set meter. So, all of the tools you pick up along the way should be looked at as such, tools, not "rules."