Imagine squeezing a tube of toothpaste and watching the paste ooze out. Notice how it takes on the circular shape of the nozzle. If you wanted to, you could keep on squeezing until the tube was empty, and then you'd have a really long coil of toothpaste, with the same circular profile throughout its length. That's all the extrusion method is: a process of squeezing material through an opening to produce a long length with a uniform cross-section.
Of course, working metal is considerably more complex than squeezing toothpaste, but the principles are the same. A billet of material is placed in a cavity with a die at one end. The die has an opening cut into it in the shape of the profile that's to be extruded. If the objective is to extrude a length of "U" shaped channel, then the die will have an opening in the shape of the "U". At the opposite end of the cavity a ram squeezes the metal, pushing it through the die.
It takes a great deal of force to push solid metal through a die but if it's heated up close to its melting point it deforms more readily. This is known as hot extrusion. It needs less force to extrude the metal when it's hot but the "extrudate," (the technical term for the material that has been extruded,) is not as strong as when it's been extruded cold. Hot extrusion also tends to wear the die faster than does cold extrusion and benefits from the addition of lubricants to help the forming process. Deciding whether to extrude hot or cold involves considering several factors such as the material itself and the end purpose or application.
Most metals can be extruded, although the most common are probably aluminum and copper. It is possible to extrude a wide range of non-metallic materials too. Plastic and rubber extrude easily, as do ceramic materials. Even pasta can be extruded!
The applications for extruded materials are limited only by your imagination. Window frames are made from extruded aluminum sections, ceramic pipes are extruded and so is the rubber of the windshield wipers on your car. Product designers like to use extruded sections because they can be a cost-effective way of producing complex assemblies.
The big advantage of extrusion is that it can produce sections that would be difficult and expensive to make by other methods. Forming the metal to the desired profile lessens the need for slow and expensive machining operations and leads to cheaper products. However, the dies are expensive, so extrusion only comes in to its own when significant quantities of a given cross-section are needed.