Router
A router is a networking device that ... well ... routes.
Its job is to take data coming in one of its ports and route it to one of the devices connected to one of its other ports.
A router is actually considered an intelligent device, as it can inspect the data and make changes to it, such as performing NAT, or Network Address Translation, that allows multiple computers to share a single internet connection and internet address.
You'll note that nowhere have I used the word wireless. The functions of a router actually have nothing to do with wireless networking.
Access point
An access point provides wireless access to a network.Honestly, that's all that an access point does.
In a sense, it's very much like a hub in that it pays no attention to the data that crosses it - it simply sends everything that it receives on a wired connection to the wireless transmitter and everything that it receives wirelessly is sent to the wired connection.
And you'll note that nowhere have I used the word router. The functions of an access point are completely unrelated to that of a router.
Wireless router
Here's where the confusion arises.Because it's such a common configuration to have both a router and a wireless access point, many equipment manufacturers have devices which do both.
But be clear - all that it really amounts to is two separate devices - a router and an access point - in one box.
That box is typically referred to as a wireless router because it's a router with a wireless access point.
Unfortunately, it's also often referred to as just a router. By now, you should see that that's technically incorrect. At best, it describes only half of what's inside a wireless router and completely confuses the situation when you don't know which of the two that you're actually referring to.
Why it matters
If you have a router - either wired or wireless - and you want to add or extend a wireless network, you don't want another router.You already have a router.
What you need is simply another wireless access point.
Here's the really confusing part: you could use a wireless router instead of a wireless access point. It would actually work - sort of.
The problem is that the router would, among other things, act as a firewall preventing machines on one side of it from seeing machines on the other side. It would also cause what's known as "double NATting", where the technique that a router uses to allow multiple computers to share a single IP address would happen twice; once for each router.
Some things would work, others would not.
Now, some wireless routers can be dumbed down to act pretty much like access points. Whether that's possible and what collection of options you'd have to select will depend on the specific model of wireless router you're working with as usual.
If you're just extending your wireless network or adding wireless capabilities to your existing network that has a non-wireless router, an access point is what you want.