In order to understand this, it is important to define some basic concepts:

Pressure

The force applied to a unit area of surface.
or rephrased,
The force (per unit area) exerted on a surface by the gas (e.g.air) or liquid (e.g.water) in contact with it.

Therefore, we can paraphrase the above explanation of ‘pressure’ to define both air and water pressure:

Air Pressure The force (per unit area) exerted by the weight of air above an object.  
e.g. the higher a person goes above sea level = less air pushes down on them = decreasing air pressure.
Water Pressure The force (per unit area) exerted by the weight of water above an object. 
e.g. the deeper a person dives below sea level = more water from above pushing down on them = increasing water pressure. 
In addition, the diver will also experience the weight of air from above the water surface.

This leads on to explaining what barometric pressure is:

Barometric pressure is also known as air pressure or atmospheric pressure and derives its name from the instrument used to measure it called a Barometer.

Units of Measurement for Barometric Pressure

The International System (SI) unit of pressure is the Pascal (Symbol: Pa) which is defined as one Newton per square meter. 
However, the unit for measuring atmospheric pressure for international meteorological purposes remains the millibar (mbar or mb).
1 mb = 100 Pa = 1 hPa.

Metric units of measurement for atmospheric pressure include:

Pascal (Pa) 
Kilopascal (kPa)      Note :1kPa=1000Pa
Millibar (mbar or mb)      Note: 1mbar=1 hectopascal (hPa)
Bar (bar)      Note: 1bar=1000mbar=100KPa which is roughly equal to the atmospheric pressure on Earth at sea level ( 0.987 atm)

Equivalent units of measurement include:

Mercury:

Millimeter of mercury (mmHg)
Centimeter of mercury (cmHg)       Note: 1cmHg = 10 mmHg
Inches of mercury (inHg)

Others:

Pounds per square inch (psi or alternatively lbf/in2, lbf/in2, lbf/sq in or lbf/sq in)
Pounds per square foot (psf)
Atmospheres (atm) Note: A standard atmosphere  = 101325Pa /101.325kPa /1013.25mbar /760 mmHg /29.92 inHg /14.70 psi