Short Answer:
A hydraulic press is a means of generating mechanical advantage by using a fluid and two different sized pistons. Hydraulic presses can be used for a lot of things. In a shop it is often used to press metal pins into holes or bearings into housings so that metal parts can be made to stay together without fasteners or threads. In a lab hydraulic presses are often used to compress powders into solid cylinders or tablets. Sticks of chalk used on blackboards are made this way as well as aspirin tablets and other common products. Hydraulic presses can be run manually using the power of someone's arm or they can use an electric pump to create the necessary pressure in the hydraulic fluid.
How it works:
If I have a piston that is three inches in diameter sitting in a cylinder and I pump fluid in the bottom of it that piston will rise and lift whatever is above it such as a 2000 pound car. The pressure of the 2000 pound car on a three inch diameter piston produces about 261 pounds per square inch of pressure in the fluid below the piston.
Piston Area = Pi*r^2 = 3.14 * 1.5 * 1.5 = 7.65 square inches
Pressure = Force / Area = 2000 pounds / 7.65 square inches = 261 pounds per square inch
If I have a pump with a piston that is only 0.25 inches in diameter (area = 3.14 * 0.125^2 = 0.049 square inches) I only need to push on it with about 98 lbs to lift a 2000 lb car.
This works out to about a 20:1 mechanical advantage. The downside is that for every inch I want to raise the car I have to push the small piston in about 20 inches. This is how a hydraulic jack or "bottle jack" you'd find in a garage works.
A practical jack or hydraulic press would have three valves and a fluid reservoir in addition to the pump and lifting piston. The pump would push fluid through a check valve, which only allows fluid to through one way, into the lifting piston cylinder causing it to rise when the handle is pressed down. When the handle is lifted the pump would suck fluid from the reservoir through another check valve. This lets you move the pump back and forth to get 20 inches or more of motion without having a 20 inch long pump piston. The third valve would let the pressurized fluid back into the reservoir when you wanted to lower the car. The reservoir would not be pressurized.
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