One standard atmosphere is exactly 1.01325 bar. The two units are close enough to be used interchangeably in casual conversation and different enough that scientific work has to pick one — which is why the atmosphere has largely lost.
About the units
The standard atmosphere was defined in 1954 as exactly 101,325 pascals, a value chosen to match the pressure exerted by a 760 mm column of mercury at 0 °C under standard gravity. That definition ties it to the older mercury-based measurements and is the reason for the odd number. The bar, by contrast, was defined as exactly 100,000 Pa specifically to be a round figure. Neither is an SI unit. The atmosphere is now deprecated in most scientific contexts, and IUPAC has since 1982 recommended the bar as the standard state pressure for thermodynamic data — a change that quietly shifted published values in chemistry tables.
The exact factor
The factor is exactly 1.01325, since both units are exactly defined in pascals. The 1.325 per cent difference matters more often than it looks. Thermodynamic data published before 1982 uses one atmosphere as the reference state and data after it uses one bar, so standard enthalpies and entropies from different eras are not directly comparable at the precision chemists work to. Real atmospheric pressure at sea level varies from about 0.95 to 1.05 bar with weather, so the standard atmosphere is a definition rather than a measurement.
Where you meet this conversion
Diving, where depth is often expressed in atmospheres absolute: every 10 metres of seawater adds approximately one atmosphere, so a diver at 30 m is at 4 ata. Chemistry and physics use it in older literature and in gas law problems, where the ideal gas constant is often quoted in units matched to atmospheres. Pressure vessel ratings, autoclaves and industrial gas cylinders sometimes carry it. In everyday speech it is used loosely to mean the bar, which is harmless outside technical work.