One bar is the pressure of 1 million dynes per square centimeter.
The pressure of one atmosphere is 1,013,000 dynes/square centimeter, which converts to 14.696 psi. The mercury-column barometer reads 760 mm in height at one atmosphere of pressure. A water depth of 33.91 feet has a pressure of one atmosphere (14.696 psi.) Each aded 33.91 feet of water depth adds another atmosphere.
...you think? It should be perfectly clear. Maybe you are having trouble with your metic units. A dyne is the unit of force in the cgs (centimeter-gram-second) metric system. A mass of one gram, at rest at sea level, experiences a gravitation field of 980 cm/sec squared. The force on the one gram mass is therefore 980 gram-cm/sec squared. A gram-cm/second squared is a dyne, so the weight of one gram is 980 dynes. A cubic centimeter of water has a mass of 1 gram, and a weight of 980 dynes. If we had a column of water 1 cm on a side, that was 102 cm high, it would weigh 1,000,000 dynes, and the water pressure at the bottom of the water column would be 1,000,000 dynes/sq centimeter, or one bar. That is almost one atmosphere, which is 1.013 bar.
OBTW, a lot of the confusion revolves around the British engineering system's use of the pound as both a unit of force and a unit of mass. A mass of one pound weighs one pound. These two units are actually called pound-mass (lbm.) and pound-force (lbf.) To convert one lbm. to .lbf, multiply by 32.2 feet/sec. squared.
Metric doesn't do this, in the standard system, (SI) the unit of mass is the kilogram, and the unit of force is the Newton. One kilogram weighs 9.8 Newtons. One gram weighs 980 dynes. One pound weighs 1 pound. Clear now?
By training and career I'm an electrical engineer. However, most of this is freshman physics, so any math major would know as much- assuming they didn't sleep through the lecture. My familiarity with British Engineering System is through one thermodynamics course of many years ago, and that is one convoluted system of units. Metric (SI) is a much easier system, although it's French, and of course we don't approve of them these days