I'll assume that you are referring to how the fuel injection system operates when it runs open loop vs. closed loop.
In brief, a modern car's engine has a computer that controls how much fuel to deliver to the cylinders via the fuel injectors. It takes into account many things including the temperature of the air and engine, the speed of the engine, the load on the engine, etc. Based on all these inputs, it checks a table that was created by the designing engineers and determines how much fuel to deliver. However, the table cannot possibly take into account all the variables that exist in the real world. So the system is designed to measure the amount of oxygen left over in the exhaust using an O2 sensor. This information is used by the computer to make small corrections to the values from the table. The goal is to get the best possible combination of fuel economy, performance, and emissions. When this is occurring, the system is said to be running closed loop.
There are times, such as when the engine is cold or when the driver floors the gas pedal that the system is designed to only use the information from the table and to ignore the O2 sensor. Under these conditions the system is said to be running open loop.