static or dynamic IP address

Did you install the server or desktop version?
Is there a particular reason why you install the rather old version of Ubuntu 10.04?
The support for desktop has ended already and for servers it will end in 7 months.

Static IP addresses should be considered for servers and devices like printers, cameras or network attached storage.

Side note, ensure that your router's DHCP server starts ip address assignments at higher addresses than your devices to prevent duplicate IP addresses.

Although I have received a couple of replies to my question, no one answered it.
(An <answer> might take the form of suggesting that I download certain software, for example.)

As to why version 10.04 was installed, the answer is interesting, but not really relevant.
My computer recently suffered a HD problem. The service technician needed a working version of
Linux on the new HD in order to proceed further. He installed Ubuntu 10.04, presumably because
he had used it before, and had the CD at hand. (Like so many computer repairmen, he specializes
in hardware, and is not expert at software. At least he knew how to use Linux. My previous service
technician needed my help to install a driver for a new video card!) But none of this is really relevant.

Can anyone suggest how I might gain the ability to use Static IP addresses?

Leslie

Talking about relevant things:
Unluckily you did not give us the relevant information in the first place and still, after I requested additional information, there is still one question open:
Did you install the server (runs in text mode) or desktop (comes with graphical login) version?

This is very relevant, because
a) the way of configuring a network interface differ significantly
and
b) the desktop version is not supported anymore and that means: no updates, easy installation of software and no support.
You will need to upgrade to a newer release then, preferably Ubuntu 14.04 LTS.

It must be the desktop version, because it has a graphical login.
I previously said that it was version 10.04, but that number is only from
my memory. Can you also please tell me how I can get the system to tell
me what the version number is? (Ubuntu seems to be very different
from the distributions that I have used in the past.)

Leslie

You can find the version number on the login screen in the most recent versions.
If not, you can open the Terminal (press Ctrl + Alt + T) and enter the command

lsb_release -a

I confirm: my ubuntu distribution is the desktop version 10.04

Can you now answer my original question?

Leslie

Hard, like many of the people who do the voluntary support here, I don't have a 10.04 desktop anymore, just because it is not supported anymore. So I refer to this video for Ubuntu 9.04. The process was very similar in the two versions:
https://www.youtube.com/watch?v=oVniJxXU89o

And as a general reminder: It is neither helpful not very clever to start a relevance discussion when you are ask for additional information. The chances that you get what you a) want and b) actually need are rather few then.

Talking about what you need: You need to update your Ubuntu installation!
For 12.04 LTS supported until 2017: https://help.ubuntu.com/community/PreciseUpgrades
Maybe also to 14.04 LTS, supported until 2019. In this case a backup of your data and a fresh installation would be best.