This measurement, typically seen in millibars or inches of mercury, describes the weight of the air above a certain area. Essentially it is how hard the air presses down from above. How does this relate to weather? Typically a low-pressure area is associated with stormy conditions. The low pressure is caused by air rising and then cooling. During that process, water droplets with condense and form clouds. The opposite happens for high pressure conditions, which often accompany fair weather.
Weather reports of barometric pressure have compensated for altitude by formulating what the local pressure would be at sea level. This way barometric pressure is measured on the same scale across the country, and altitude can't cause deceptively low pressure readings.
Most weather stations give barometic pressure in inches of mercury, or "inHg." Standard sea-level pressure is 29.92 inHg. Many meteorologists also refer to pressure in millibars (equivalent to hectopascals, hPa). The sea-level pressure in millibars is 1013mb.
Visibility is simply a measure of how far an object can be clearly seen for given weather conditions. How big of an object? It depends on the distance. The idea is to measure how much the atmosphere affects visibility, not how well your eyes work! On the clearest days visibility will usually be listed as "unlimited," although that usually means around 50 miles or so.
Foggy or rainy weather may reduce visibility to only a few hundred feet. Wikipedia says the international definition of fog is visibility less than 1km, mist is visibility from 1 to 2km, and haze is visiblity from 2 to 5km.