Will all these computers have the same external IP address, or will it somehow be different for each one? And does it matter?
These computers, behind the router, will appear as just one computer from the outside (at least they will use the same IP address; websites are still able to distinguish them by using browser cookies and some information that the browser sends to the server).
Generally it doesn't matter, but all limitations which are IP-based apply. For example, if your IP is banned from editing some site, all of the computers behind the same router will be banned.
What is Static IP
Actually, you should distinguish between static vs dynamic IPs (it's a method of assignment), "real" IPs vs IPs from the private ranges, and internal vs external IP addresses.
Static means that you insert the IP address in the network settings directly, it is pre-assigned to this machine, and it is always the same. The machine doesn't need to ask anybody to know its own IP address.
Dynamic IP means that you do not assign any particular address in the network settings. Instead, it is assigned automatically (via DHCP). For this automatic configuration to work, there needs to be a device on the network, which gives IP addresses. This is called DHCP server. Usually, it's home router's task.
It doesn't matter in what way the address (and other network settings) were chosen, as long as they are correct. These are only two different ways to configure the network.
External and internal IPs
There are not enough possible IPv4 addresses available. To workaround this limitation, addresses from the private ranges (10.x.x.x, 172.16.x.x, 192.168.x.x) are used in the internal network, and the "real" public IP is used as a single gateway for the whole network. This is called Network Address Translation. Most home routers do this.
( Internet, outer network )
188.8.131.52 (external IP)
[ NAT device ]
192.168.1.1 (internal private IP)
[ PC of Alice ] [ Bob's laptop ]
It doesn't matter if your external IP is assigned statically or not (usually it is dynamic, given via DHCP, even if it appears to be the same most of the time), it doesn't matter if your internal addresses are assigned statically or dynamically. The network works as long as it is configured consistently, and all computers know that they should send everything to the router, and the router knows how to rewrite address for the outer network.
You will appear as using 184.108.40.206 IP address to most of the sites (you may go to ipchicken.com to check). And if you don't have Java installed, they will not be able to say what is your internal address (but Java plugin may report it; probably Flash and Silverlight can do it too).
It may happen, that the outer address of the router is from the private range too (e.g. 10.1.1.1). This means that there is at least yet another NAT before the internet.
When a static "real" IP is necessary?
There is a situation, when you want to have your external IP to be static and not from the private range ("real" IP). Usually, this is the case when you run servers at home.
You don't want them to change their IP addresses spontaneously (but you can use dyndns to deal with it). And you don't want your server to have an address from the private range, because then nobody from the outside can connect it. If it uses a private IP, you need to configure routers all the way to the outside to make server connectable from the outside. If your router's external address is not from a private range (is a "real" IP), you need to configure only your own router. But if router's external address is from a private range, you depend on your ISP to configure their routers for you (good luck with that).