Bicycle pumps work through the manipulation of air pressure with a piston inside a cylinder. When the lever is pulled up, the air is drawn into the cylinder. When the lever is pushed down, the air is expelled, usually into the tire, assuming it's attached. The system of one-way valves ensure this happens correctly.
Most floor pumps generally have a gauge to allow the monitoring of air pressure within the tire as it's inflated. Various tires have different pressure maximums, although tires with Schrader valves (the thicker ones) max out at about 50 psi for mountain tires and 70 psi for hybrid tires, while Presta valves go higher for road tires, up to about 130 psi. When a new tire is bought, the packaging will usually indicate the air pressure limits for that tire, but when the bicycle already has tires on it, the air pressure maximum appears on the side of the tire. Going outside the suggested range of the tire often leads to tire damage, often at the worst time, with blowouts happening during a ride. Using tires designed to avoid flats and tubes that resist punctures make it easier to keep going without a flat. Riders should keep a hand pump attached to the bicycle frame or a couple of CO2 cartridges to inflate road tire tubes to full pressure easily.