This is a short explanation on the difference between the three items in the title.
When you get a connection from your internet service provider (ISP), they will almost always include a 'router'. This is a device which gets you onto the network and allows other devices within your home to also connect. Many of these routers will also do WiFi. In actual fact, routing is only part of the story. If your 'router' connects to the internet via ADSL, Fibre, Cable (Virgin etc) then it is actually a modem and router in one box. If it does WiFi as well, it is a Modem, Router Wireless access Point. So what are all these devices?
Modems
Your ISP will supply you at least with a modem. This will connect to your phone line, incoming fibre internet cable or coax Cable internet cable. It will do a Point to Point Protocol (PPP) connection to your ISP. Now you could take the output of a modem and plug it into a PC (or Mac) and with the correct settings you would get connected to the internet. However, you would only have this one computer connected.
If you are a bit of an expert, you could set your computer up as a router and have additional devices connect via this. However, this is very complicated and unless you really know what you are doing, it is a major security risk. You computer would be directly connected to the internet with no firewall or safeguards.
What you could do is connect the output of a modem to the input of a router. This is not done much in residential environments as most domestic routers are actually modem/routers and the modem is built in.
Draytek 130 ADSL VDSL modem
As you can see, this has an input for the phone cable and an ethernet cable out for the WAN port of the router.
So why would you use a modem? If you were running a commercial operation, you may have multiple internet connections and a router with multiple WAN ports. Each one could be fed via a modem. Draytek 2862 routers have an ADSL port, with inbuilt modem plus a WAN port for a modem which can be used for a second internet connection. They also have USB which can be used for a mobile data backup as well.
Router
A router takes the output of a modem into its WAN port (Wide Area Network). It then connects this to all the devices on the network. So how does it do this all work? The first thing is Dynamic Host Control Protocol (DHCP). This is simply the way the router hands out IP addresses. When a device connects to the network, it will send out a broadcast message asking if there is a router available to provide an IP address. If there is, the router will respond with three things:
IP address - this is the local address on the network and looks something like 192.168.1.10
Subnet Mask - This tells the device how big the network is. The device uses this to determine if it is trying to connect to something on the same network or on a different one. If it is on a different network, the request is sent to the Gateway.
Gateway - This is normally the address of the router and if the required destination lies outside the local network, the information is sent here to be forwarded on.
DNS - Computers speak with IP addresses but humans find these difficult to remember so the DNS system translates between domain names, such as www.bbc.co.uk and IP addresses 212.58.233.254.
routers also do lots of other things, such as NAT (Network address Translation), DYNDNS, VPN and many other functions.
Wireless Access Point (WAP)
A WAP is a simple device for connecting wifi devices to the network. Many routers will have built in WAPs but some will not. If you router is in a basement plant room, there may be little point. If you have ethernet cables from your router run around your house, a good option for extending your network is to use WAPs. These don't extend or relay the wireless signal, they create a wireless network. If all the WAPs have the same SSID (wifi name) and password, it will appear as if the network runs throughout your house.
Something to be aware of is that many devices will hold on to their connection to a WAP, even if they move closer to another. So if you are in the bedroom and you walk downstairs, you may appear to be getting a very poor wifi signal, despite there being a second WAP downstairs. If you disconnect from the network and reconnect, you will probably connect to the stronger signal. To get around this, you need a managed wireless network.