Verify adapter plugged into a wall outlet
Make sure the AC adapter is firmly and securely plugged into the wall outlet. This may sound obvious, but many times the adapter can become loose without even knowing. If it's not plugged in all the way, it can make it seem like the adapter is not working, but it may just not be getting power from the wall outlet. When in doubt a good way to make sure it is plugged in properly is to disconnect and then reconnect the connection.
Check adapter indicator light
Many laptop AC adapters have a power indicator light on the adapter brick. Check for this indicator light and if there is one, make sure it is light up with a green light. This means the adapter is at least receiving power.
Check the wall outlet itself
Some wall outlets require a light switch to be in the on position in order for the outlet to be active. Many newer houses, apartments, and condos feature this type of wall outlet. If the light switch is on, verify the wall outlet is working in general. Use a volt meter to measure the power output and verify the outlet is functioning properly. Alternatively you can also try another wall outlet. Also, double check the circuit breaker and make sure nothing is tripped or turned off.
Check adapter brick connection
Most laptop AC adapters come in two pieces. Make sure both of those pieces are connected firmly and securely. The two pieces connect at the rectangular box piece, often called the brick. This connection can become loose or disconnected relatively easily, so want to make sure the connection is good.
Adapter plugged into laptop
This may seem obvious as well, but the adapter can become disconnected from the laptop pretty easily. The metal tip on the adapter cord needs to be firmly attached to the laptop in order for the laptop to receive power from the adapter. Also, in either the metal tip on the adapter or the connection on the laptop, there is often a small metal pin. Make sure this pin is not bent or broken as that can cause a problem with the laptop receiving power from the adapter.
Laptop power connection port
This is a bit more difficult to check or diagnose. The power connector is connected directly to the laptop's motherboard, usually through one or more solder points. If a solder point is damaged or has broken away from the motherboard, this will prevent the laptop from getting power from the AC adapter. Fixing this requires very precise work with either soldering the power connector back onto the motherboard or replacing the power connector itself, if it is defective. In some cases, this kind of problem is not repairable and will require a replacement of the motherboard or power board.
If you think this may be your problem, it is something that's best left to the expert technicians who may need to take apart the laptop to get the motherboard or power board and diagnose the problem.
If you've determined the AC adapter is not working properly and needs to be replaced, you can try to get a replacement from the laptop manufacturer, which will usually sell an exact replacement or compatible universal AC adapter.
You can also opt to purchase a universal AC adapter from a local or online electronics retailer. One brand to look for is Targus, which is a quality brand and a popular choice. Targus makes several universal AC adapters which work for many laptop brands and models. You will want to make sure you purchase one that matches the specs of your existing AC adapter, including the voltage and amperage. Also, check the packaging to make sure it will work for your laptop brand and model.