These are similar but not identical concepts. Rhythm refers to the overall tempo, or pace, at which the poem unfolds, while meter refers to the measured beat established by patterns of stressed and unstressed syllables. Poets who write free verse, generally de-emphasize or ignore meter and focus instead on refining and tuning their natural speech rhythms to suit the poem's tone and content. Or as Ezra Pound put it, they "compose in the sequence of the musical phrase, not in the sequence of the metronome."
Still, even if you write mostly free verse, understanding some basic metric principles can help. As has often been pointed out, English sentences naturally tend to establish a dominant beat, usually iambic, so if you have a troublesome line or phrase that "just doesn't feel right," you may find that by quickly scanning your line, you can spot and fix the problem, replacing a one syllable word with a two syllable word, or vice-versa. Besides helping with such quick rhythmic tune ups, metric awareness is essential for writing in traditional fixed forms, such as blank verse or the sonnet.
Here, very briefly, is how to scan a line. First read it aloud to get a feel for where the stressed and unstressed syllables fall.
When Ajax strives some rock's vast weight to throw,
The line too labors, and the words move slow;
--Alexander Pope, "An Essay on Criticism"
Then mark the syllables as being stressed or unstressed. Mark unstressed syllables with a 
and stressed syllables with a 
.
Next, look for patterns. First check the syllables two at a time; then if no pattern is evident, three at a time. When you see a repeated pattern, use a / to divide the line into feet.
If you're in doubt, try another line:
This second line is more difficult, partly because of the three lightly stressed syllables " . . . bors, and the, . . ." Yet this very faltering, laboring for consistency seems to emphasize Pope's points about effort and struggle, about sound echoing sense.
By now we can recognize a dominant meter:
And knowing that meter helps us fit the doubtful syllables into the established pattern, in this case iambic pentameter.
This short discussion just gives the barest essentials. How deeply you get into metrics depends upon your level of interest and the type of poetry you write.