Amigurumi is the art of crocheting or knitting stuffed animals and anthropomorphic creatures. The word amigurumi is a combination of two Japanese words: amu (to knit) and nuigurumi (stuffed toy). Most common themes include: animals, creatures, food, and people.
I've been following Amigurumi for several years now. I don't know the exact origins of this craft and I don't think anyone really knows for sure. But I've watched many amazing designers come and go in the past years and they've inspired many people to create their own amigurumi.
But the old amigurumi designs from about 5-10 years ago? You're going to find them on personal Japanese blogs/sites. If you search "編みぐるみ" (amigurumi) on Google, you just might find some that are still up today.
Now there are many different styles of amigurumi. I believe amigurumi can be made in which ever techniques to create the look and shapes you want. Whether it is in the round, flat, worked in spirals, joined rounds, any kind of stitch (even the tall and unique ones), crocheted, knit, etc.
Materials that are commonly used to make amigurumi include: yarn, stuffing, plastic safety eyes, and felt for facial features or embellishments.
Tools: crochet hook, knitting needles, yarn needles, embroidery needles, and scissors.
Common skills: crochet, knitting, math, sewing, and embroidery.
There are several other materials & techniques people use including: fabric to create clothes and accessories, glue instead of sewing on pieces, making some pieces out of polymer clay, buttons, beads, sequins, needle felting, and more!
It's important to experiment to understand the construction of crochet or knit pieces. You may need to brush up on your math skills because it'll probably help you when you need to create pieces in a certain size! Just make sure to keep notes of what you do and use so you can refer back to it later.
If you're interested in learning how to make amigurumi, there are many patterns and tutorials on the internet.