Lizat wrote

What I think happens with Powerline is that instead of using ethernet cabling it uses the power network to carry the signal. 

  @Lizat 

That is correct.

Lizat wrote

But what is the difference between an adapter and an extender and under what circumstances are they used? Are there performance implications, or other pros and cons? 

Generally, there are two types of Powerline adapters for home use.
There are simple Powerline adapters that only provide Ethernet port(s) for connecting network devices via Ethernet cable. 
And then there are Powerline adapters that additionally contain a built-in Wi-Fi AP, allowing client devices to be connected via Ethernet cable as well as wirelessly. I guess this type of Powerline adapter is often called "Powerline Extender" because it can be directly used to "extend" the Wi-Fi coverage.
You can distribute Powerline adapters of both types throughout your house according to your own needs. At least two such "units" are required to create a Powerline connection, but more can be added if needed.

The thing with Powerline technology is that the achievable data throughput heavily depends on the properties of the building's electrical wiring, the distance between two Powerline adapters and also the amount of electrical interferrence within the electrical wiring of the building. It is nearly impossible to predict the resulting data throughput in advance. (I mean, before the purchasing the Powerline adapters)
If you are really lucky and the circumstances at your home are favorable, then you might end up with 200 Mbps speed, but if you are not so lucky, then you could also end up with only 20 Mbps. In worst case you could even end up with no working connection at all.

Lizat wrote

I read that there are access points as well, so what are they and what is their use?

An Access Point is usually connected to the main router directly via an Ethernet cable and then distributes this network connection via Wi-Fi to nearby wireless clients. Unlike the Powerline technology, an Ethernet connection should provide the advertised speed up to a distance of 100 meters. (or more precisely, 100 m cable length)
Of course, the difficulty with this solution is to get the Ethernet cable installed in the house, between the main router and the access point.