Going up a dimension, here's a 3-D vector function:
f(t) = <cos t, sin t>.
For 0 < t < 4π, what does the graph of this function look like?
If we didn't have the third coordinate to worry about, this would be the unit circle again. Now it's like the unit circle, but each point is higher than the point before it:
It looks like a slinky or a blood-draining helical turn on a roller coaster.