The oil filter in your car's engine filters some of the dirt and particles out of the oil in order to keep it a bit cleaner so that it is better able to lubricate your engine. Some people like to run their oil (especially synthetic) for extended intervals before they change it, but find that while their oil is still good, the oil filter is not up to the task. There are sometimes different oil filters available that are able to hold more dirt, but in some cases you are stuck with the stock filter. If this is the case, or if you want to use a specific filter, it is possible to change the oil filter without changing the oil.
Before you start ensure that you have the proper filter for your vehicle, a pan to catch any oil that leaks out, and some fresh oil to top up the system. Depending on the vehicle this process may result in quite a bit of spilled oil or only a few drops, but normally you should expect to lose around half a quart.
Unlike a normal oil change where you start with a warm engine to allow the oil to pour out more easily, when changing just the filter its best to start with a cold engine. With the vehicle safely supported (if necessary), place the catch pan under the oil filter. Remove the old oil filter, being prepared for some of the oil oil to leak out. Use a rag to clean up the old oil as well as you can. Prepare the new filter by applying some fresh oil to the seal, and if possible (depending on its orientation) filling it with oil. Screw the new filter onto the engine and snug it up by hand. Check the oil level and top up as necessary.
Once you have verified the oil level start the car and check for leaks around the new oil filter. Take the car for a short test drive and then again check the oil level and check for leaks.