The Bernoulli effect is based on the conservation of energy applied to fluids. The kinetic energy density plus the pressure (plus any potential energy) at one position is equal to the same sum in a different position. High velocity fluid will have lower pressure than low or no velocity fluid.
Blowing air through a pipe with a narrow section in the center (called a Venturi) demonstrates the Bernoulli effect in the most rigorous way. The smaller center pipe causes the air to speed up. The higher kinetic energy produces a lower pressure. This is observed as a change in height of a fluid (blue) connected to the Venturi on one end and to the outer pipe on the other end. Without the air flow, the height of the fluid equalizes.
Use a straw to blow air between two aluminum pop cans or two Styrofoam cups hanging by threads. The larger kinetic energy of air between the objects lowers the pressure. The objects move together.
Place a spool of thread on top of a 3 x 5 card. Blow air into the top of the spool. The air rushing out the bottom produces lower pressure on the card, such that the card can be picked up using only this airflow. This has been commercialized for the movement of small delicate parts. See the picture to the left.
Hold a sheet of 8.5 x 11 paper by one end under your lower lip. The paper should curve over your fingers and hang downward. Now blow over the top of the downward hanging paper. The larger kinetic energy of the air over the top reduces the pressure. The paper rises up due to the greater pressure in the still air underneath.
Hang a small piece of packing popcorn in front of a loud speaker. Hook a signal generator up to an acoustic amplifier. The amplifier connects to the speaker. Set the frequency to around 100Hz and turn up the amplitude. The popcorn will move towards the speaker. The amplitude of the sound wave is larger near the speaker. Thus the movement of air is larger and thus the kinetic energy. The average pressure is lower near the speaker than it is further away.