One bar is 14.5037737730 psi. The bar is defined as exactly 100,000 pascals, and the psi follows from the pound-force and the square inch, so the relationship between them is exact even though the number is untidy.
About the units
The bar was introduced in 1909 by the Norwegian meteorologist Vilhelm Bjerknes, who wanted a unit close to average atmospheric pressure at sea level. He got close: one bar is 0.987 standard atmospheres, about one per cent below. It is not an SI unit and the SI authorities discourage it, but it survived because 100,000 Pa is a convenient round number and because atmospheric pressure being roughly one bar makes gauge readings intuitive. The psi — pounds per square inch — is the corresponding unit in the imperial world, defined as one pound-force acting on one square inch. Note that pound-force is a force and not a mass, which is where the unit differs fundamentally from the pound used for weight.
The exact factor
One psi is exactly 6,894.757293168361 Pa, derived from the pound-force of 4.4482216152605 N divided by the square inch of 0.00064516 m². Dividing 100,000 by that gives 14.5037737730209 psi per bar. The everyday approximation of 14.5 is accurate to a quarter of a per mille, which is far tighter than any tyre gauge. A useful mental anchor is that 2 bar is about 29 psi and 2.5 bar is about 36 psi, which brackets nearly every passenger car tyre.
Where you meet this conversion
Tyre pressure is where most people meet it. European car door plates specify bar while American gauges read psi, and a hire car crossing a border presents both. Bicycle tyres run far higher — a road tyre at 8 bar is 116 psi — and getting the unit wrong there is dangerous rather than merely inconvenient. Diving equipment uses bar throughout, with a full cylinder at 200 or 300 bar. Hydraulic and pneumatic systems, espresso machines and pressure washers all specify in one unit or the other depending on where they were designed.