Definition
Consonants are sounds made when the flow of air is blocked or squeezed in the throat or mouth. Vowels are sounds that flow out smoothly through the throat and mouth with nothing in the way. Together they are the most basic building blocks of speech. A vowel can form a syllable (one beat of sound) all by itself, but a consonant needs a vowel to make one.
The sound of pinching a hose: consonants
When you water the garden with a hose and pinch the end with your thumb, the water shoots out hard and makes a loud hiss. The water was flowing quietly, but now it crashes into your thumb and scatters.
Consonants are made in exactly the same way. (In Korean, they are called "닿소리" (datsori, "touching sounds").) Breath pushes up from your lungs and passes through the throat, the roof of the mouth, the tongue, the gums, and the lips. Somewhere along the way it bumps into something or the path gets narrow, and that makes a sound.
For example, to say the Korean letters "ㅁ" (m) or "ㅂ" (b/p), you press both lips shut and then release. To say "ㅅ" (s), you bring the tip of your tongue close to your upper gums and push air through the narrow gap. Sounds that get in the way of the airflow like this are called consonants. They include stops (block, then burst), fricatives (air squeezed through a gap), and nasals (air out through the nose: "ㄴ" n, "ㅁ" m, "ㅇ" ng).
Breath that rings out freely: vowels
When you ride a thrilling roller coaster or reach the top of a mountain, you shout "Wow!" or "Woo-hoo!" without thinking. You open your mouth wide and let the sound pour out.
Notice what your mouth is doing. Your tongue and lips aren't forcing the breath shut. Air vibrated by your vocal cords slips smoothly out through a wide-open path.
You can change the sound just by rounding your lips or moving your tongue up and down, without ever blocking the air. Sounds like "아" (a), "오" (o), and "우" (u) that ring out with nothing in the way are vowels. (In Korean, they are called "홀소리" (holsori, "standing-alone sounds").) Because nothing blocks them, vowels sound clear and carry far. They form the solid backbone of speech.
A bit more precisely: a mother and son in harmony
A bit more precisely, the Korean names for these sounds hide a story about how the two relate. In Chinese characters, the word for consonant (子音) means "son sound," and the word for vowel (母音) means "mother sound."
The mother sound, the vowel, can stand on its own. That's why a single vowel like "아" (a) or "이" (i) can carry a clear sound by itself. The son sound, the consonant, can't ring out long and clear alone. It has to be held in the mother's arms, the vowel, to become a complete syllable.
For example, the Korean letter "ㄱ" (g/k) by itself chokes off the breath and is hard to hold. (Some consonants, such as "ㅁ" m and "ㅅ" s, can be stretched out, but even they can't form a syllable alone.) Add "ㅏ" (a) after it, and you get "가" (ga), which rings out strongly. Hangul, the Korean alphabet created by King Sejong, builds in this principle with amazing order. The consonant letters are designed to copy the shapes of the speech organs, like the tongue and lips. The vowel letters are built to symbolize the harmony of heaven, earth, and humans. When the two meet, they form one perfect syllable.
Common misconceptions
A single consonant letter can be held for a long time as a complete sound on its own.
Stop consonants like "ㄱ" (g/k) and "ㅂ" (b/p) can't be stretched out alone. Even consonants you can draw out, like "ㅁ" (m) and "ㅅ" (s), can't form a syllable by themselves in Korean. To make a clear syllable, they need a vowel.
The Korean letter "ㅇ" always makes the same consonant sound, wherever it sits in a syllable.
At the start of a syllable, as in "아이" (ai, "child"), "ㅇ" is just an empty placeholder with no sound. Only at the end of a syllable, as in "강" (gang, "river"), does it become a real consonant: the back of the tongue blocks the soft palate and the sound comes out through the nose (ng).
Where you meet it
When you hold a high note at karaoke, you stretch a vowel like "아~" (a) or "이~" (i), not a consonant.
When a baby babbles, vowels like "아" (a) and "우" (u) come first. As the baby grows, lips start to bump together and consonants join in, as in "엄마" (eomma, "mom") and "아빠" (appa, "dad").
In a Hangul block game, you combine consonant cards (ㄱ g, ㄴ n, ㄷ d) with vowel cards (ㅏ a, ㅓ eo, ㅗ o) to build syllables like "가" (ga), "너" (neo), and "도" (do).
In one sentence
Consonants are sounds made by blocking the airflow, and vowels are sounds that burst out with nothing in the way. Together they make complete speech.