I didn't read through all the replies so I may repeat a few things, but I wanted to put in my knowledge because I love love love Illustrator. I learned the program in high school and have found a million uses for it since.
If you think about pixels as little building blocks and vectors as a ball of string, you'll understand them easily. Pixels stack next to each other to create an image, like when you see the pictures made of smaller pictures. From far away, they look like a portrait of Elvis but up close you can see tons of smaller images making up the shadows on his nose, etc. As a single smooth line, a vector can be blown up as large as you want and still be a smooth line. PDFs are a great format for saving artwork because they preserve the vector, aka smooth, scaleable lines, whereas JPG's compress the image into pixels. I think everyone is clear on that, but I wanted to repeat it just in case.
The pen tool in Illustrator lets you set anchor points (imagine using push pins to secure a point on the string of yarn) to create paths (also called strokes). The area between anchors on a path can be filled with color or left blank.
You can set the anchors yourself by tracing over an image (click to set anchors with a straight line between or click and hold while pulling the mouse to "stretch" the path into curves), or you can use Live Trace and Illustrator will look at your image and create paths for you. The important thing to know about this is you can adjust how detailed Illustrator gets in tracing for you, and figuring that out just takes some fooling around. Once Illustrator Live Traces, you can use the pen tool to delete erroneous anchor points and smooth out the path.
Say you've got a zig zagged line of yarn-- remove all the points between the first and last anchors and tug...now you have a straight line. Hopefully that makes sense! Illustrator just connects the dots, so delete the dots you don't need.
Let me know if this helps or is just confusing!