How do pick and place machines pick up components?

Using a vacuum pickup. Those are handheld ones, but the machine has one on a robot arm with XYZ control.

It's very simple, just a tiny suction cup with a vacuum that can be turned on and off.

The components are placed onto the solder paste which is slightly sticky, so small vibration won't dislodge them. If the board will be double-sided, a small dot of glue is added before the component is placed. This prevents the components falling off when the other side is being soldered.

Then the board goes through the oven, is subject to optical inspection to check that the parts have soldered correctly, and any parts that can't be handled by the pick and place machine are hand-fitted and hand or wave soldered. This is usually only required for big connectors.

They use a vacuum pickup, typically with a spring-loaded tip so that the pickup has some compliance in the Z direction. Since the force is limited by atmospheric pressure multiplied by the nozzle bore cross-sectional area, different sizes of nozzles are used for different parts.

Here are some typical nozzles used on one major brand of P&P machine:

Not shown here, but they can also be rectangular (note the keying flats to prevent rotation).

Vacum pickup is indeed the normal way. A machine will often have different nozzles for different sizes of component. A camera is often used between picking and placing to check the position and orientation of the component on the nozzle.

Connectors will often have a small plastic or metal cap or a peice of kapton tape to provide a flat surface for the vacum to pick.