Zapped pretty much explained it.
An atomizer used primarily for dripping looks like this:
Vital Vapers : eGo / kGo 510 Atomizer [510-atomizer] - $9.99.
When you look at them, the inside will most likely look like this:
http://media.photobucket.com/image/...20adds/lowbridgeandhighbridge901atomizers.jpg.
You screw the atomizer onto your battery, drip 2-4 drops of e-juice onto that metal mesh bridge, snap on a drip tip, and vape. 2-4 drops is typically good for about 4 to 5 "hits", "puffs", "drags", whatever you want to call it. When it tastes nasty (trust me, you'll know). simply take off the drip tip, drip 2-4 more drops, snap the drip tip back on, and vape! Repeat the process. This is "dripping".
Cartomizers look like this:
http://ecigarettereviewsite.net/wp-content/uploads/2012/02/e-cigarette-cartomizer.jpg, and when you look at the inside, they look like this:
Google Image Result for http://www.evcigarettes.com/product_images/uploaded_images/em-riva-510-cartomizer-inside.jpg.
Cartomizers come with "polyfill" fiber and needs to be "primed"; in other words, you need to fill a carto like a Zippo lighter, only more slowly and carefully. Once properly primed (dripping 20 or so drops directly into the polyfill, avoiding the hole in the center of the carto, while giving each drop time to soak in), you don't need to keep adding "drops" of e-juice.
I used to drip exclusively until I figured out how to properly prime and use a cartomizer. Now I use them exclusively, in combination with tanks. NOTE: If you're using cartomizers without tanks, you will eventually have to "top off" the cartomizer by periodically adding more drops of e-juice into the polyfill. A carto can last you anywhere from a week to two weeks. They're cheap, use and toss.
Atomizers are bit more expensive, but if properly used and cleaned, they can last up to a couple months.
Hope that helps.
Edit: some people prefer the "condom method" for filling cartomizers. Do a Youtube search for "cartomizers + condom method" to learn how to prime a cartomizer using this technique. Works, is quick, but very, very messy. Be sure to have a paper towel, a rag, and some hand sanitizer nearby. And make sure if you blow out the excess fluid, not to get it inside your mouth. I made the mistake of accidentally getting some 2.4 mg nic juice in my mouth, and I was pretty dizzy and nauseous from the little bit I ingested.