Capacitors have been explained as behaving like small batteries having a charge then discharging. This explanation falls quite short of the true nature of capacitors. This article will not attempt to get too technical about their function for now that explanation will be good enough.
The capacitors that cause us all the trouble are called electrolytic capacitors. They are made up of two long pieces of foil separated by a piece of special paper or aluminum oxide called a dielectric wound tightly and placed in a small aluminum can, they are also filled with an oil. Two wires come out of the bottom one connected to each separate piece of foil. If you want a complete description Wikipedia has a nice article.
Capacitors cause us issues because they can become leaky electronically and physically. If you see a puddle of brown liquid or crusted brown substance under a capacitor chances are it is failing. Circuit failures are usually a chain of events. A blown transistor is rarely the actual cause but a symptom of the problem. Don’t get me wrong a transistor can fail on its own but more times than not another root cause is suspect, usually some passive component failure like a capacitor short. Capacitor failures can have many symptoms from subtle to extreme. For example (not a complete list) system lockup, low voltage, wavy lines or distorted picture on a monitor, blown fuses, “Big Bang!” when one blows up.
In my years of experience the most likely capacitors to fail are smaller in value like 100 mfd or similar type and they are located near a heat source like a large resistor or a transistor heat sink. Large filter power supply capacitors fail less often. In many cases people will change out all the capacitors and still have issues. They will buy capacitor kits and preform “Preventive maintenance” You might fix some of your issues by doing this but it is no subsite for troubleshooting and finding the problem. Most electrolytic capacitors don’t need changed out and will work for a very long time. With one exception of the wax paper capacitors in old tube radios.
More effective in lieu of troubleshooting will be repair kits that contain the common failure components. But again this method is hit and miss.