Quick answer: Drop spindles and spinning wheels twist loose fiber into yarn. The yarn can then be plied for additional strength.
Handspinners can use a drop spindle or a spinning wheel to create yarn from unspun fibers from animals or plants. The fibers might come from cotton bolls, flax stems, sheep fleece, or hairs from animals such as goats, dogs, llamas, alpacas, or camels. Though spindles and spinning wheels look very different from each other, they both perform the same function: they twist fibers around each other into a yarn. The yarn might be very fine, like sewing thread, or it might be very thick. The yarn can be a single-ply yarn, called singles, or it can incorporate multiple plies. But no matter what kind of yarn a handspinner makes, twist holds the individual fibers together.
Once humans discovered how to twist animal and plant fibers into yarn, they began making cloth from that yarn. It's much easier to weave, knit, or crochet with yarn as opposed to a handful of untwisted fibers. Plus the resulting fabric will hold up better under daily use.
Drop spindles
People started using drop spindles in prehistoric times, because a spindle is really easy to make and use. A long thin shaft goes through a round weighted part, called the whorl. The spinner feeds unspun fiber from one hand and spins the spindle with the other.
To begin making yarn, the spinner drafts some fiber—that is, pulls out a little fiber from the fiber supply. When the spinner sets the spindle rotating, twist builds up in the drafted fiber. When the yarn has enough twist, the spinner wraps it around the shaft and drafts again. When the spinner has made enough yarn and wrapped it into a cone-shaped mass around the shaft (in what's called a cop), the yarn can be removed.

Spinning on a drop spindle
| A. whorl | 1. fiber hand releases |
| B. shaft | 2. drafting triangle |
| 3. forward hand pinches | |
| 4. Z (clockwise) twist |
Spinning wheels
A spinning wheel does the same thing as a spindle: it twists fiber into yarn, then provides a place to store the yarn. A wheel can generally make yarn faster than a spindle, however, because the spinner's feet get involved. In addition, the bobbins used on a wheel can store a lot of yarn.
As the spinner treadles, the treadles set the drive wheel in motion. The drive wheel sets the flyer in motion. The flyer provides the rotation that builds up twist, and the bobbin holds spun yarn.

Spinning on a spinning wheel
| A. treadle | 1. fiber hand releases |
| B. flyer and bobbin | 2. drafting triangle |
| C. drive wheel | 3. forward hand pinches |
| 4. Z (clockwise) twist |
Think of the flyer and bobbin as a spindle held parallel to the floor. When the spinner has filled the bobbin with yarn, the bobbin can be removed.

Today, spinners can use modern wheels with flyers and bobbins, or their ancestors and cousins, the great wheel (walking wheel), the quill wheel, and the charkha wheel. These older wheels use a sharp point instead of a flyer to build up twist in the fiber—they're even more similar to a spindle. Sleeping Beauty pricked her finger on the sharp point of a quill-style wheel. Spinners can also use electric spinners (e-spinners), where a motor turns the flyer to add twist.
Plying
Spinners create plied yarns from singles, using a spindle or a spinning wheel. The spinner fills two or more bobbins or makes more than one cop on a spindle. Bobbins go on a lazy kate, which sits on the floor or attaches to the wheel. Then the spinner rotates the spindle or the flyer counterclockwise, for S twist, until the plies wrap around each other. Plied yarns are generally thicker and stronger than singles yarns. The two colors of singles in this barber pole yarn make it easy to see the plies.

Resources
Beginning Spinning on a Wheel, online course taught by Stephanie Flynn
Beyond Your Default Spin-along, free online course taught by Stephanie Flynn