Short answer see if it is in the repos first, if it is install that version.
Long answer it depends how the package has been built.
To start the install open a terminal cd to the directory you downloaded it to.
Then tar zvxf name of package, once it has unpacked, hopefully with no errors.
The cd to the new folder created and look for a readme or INSTALL file and
read it for further instructions.
If you want it system wide then use sudo in front of the command when you
install it, whether it be make ./install etc.

That is like a zip file. You need to

http://ubuntuforums.org/showthread.php?t=768148

There is a good thread on it all.

There are instructions there that deal with build-essential,
./configure, make, and install. I'd like to play around with
this a bit, see if I can learn something -- mentioned a
sandbox in my previous post. I don't want to mess anything
up because I depend on this computer...someone nudge me in
the right direction please?

You should not be in the habit of downloading and installing programmes
from outside of the repositories unless:

a) you know what you are doing
b) you do don't care about messing up your installation
c) you don't expect others to bail you out afterwards

The best and the only way, to my way of thinking, for newbies to install
software is to use the package manager or download deb files that are
specifically made for your version. Otherwise you risk not being able to remove
it properly later and will have to live with the results.

Any idiot can post a file to the internet and any idiot can download and install
it. However, there are no guarantee that the file will work and won't mess you
up royally. People go to a lot of trouble to put packages in repositories and
can guarantee their authenticity. There are between 24 and 35000 packages in the
repos. There is very little that you might need that is not available. On top of
this you have other resources at your disposal such as getdeb.net.

There is something to be said for learning how to compile from source code, but
you should not do this on your main computer.

People can tell you how to do anything, but whether you should be following
their advice is another story.

My advice is to stick to the repositories found in Synaptic until you understand
more about how Linux works and why things are the way they are.

I'm a few steps beyond newbie level
and I know he's right. For someone looking to learn Linux
configuration or programming -- on a system that can be
restarted from scratch if necessary without losing a working
environment -- of course that's a different story. Such a
person, I'm thinking, might ask a targeted question that
includes some details beyond "downloaded a linux program"
and (someone correct me if I'm wrong) might consider
setting up a sandbox with chroot.

I'd like to learn how to set up a sandbox myself.
Could someone point me in the right direction?

I don't want to mess anything
up because I depend on this computer...someone nudge me in
the right direction please?

IMO, the best education would be to install a basic Linux distribution such as
Slackware, Arch, Gentoo or Linux From Scratch. By basic I mean that they force
you do things manually that other distro automate as part of the installation
process. That way you can see how it works under the hood. Once you have a sense
of what is going on when you do something then you can do whatever your mind
wants, including making your own distribution.

However, you mention a sandbox. For that you should have a second computer or a
drive that you don't care about. Install Linux there, but keep it isolated from
your main operating system.

http://www.linuxfromscratch.org/

That's an interesting idea, but I'm specifically interested
in chroot because I need to convert some of my thin clients
(Edubuntu 8.04 LTSP) to "fat clients" a la

https://help.ubuntu.com/community/UbuntuLTSP/LTSPFatClients

and doing it from scratch looks pretty complicated. I'm
committed to Ubuntu LTSP for now, depend on it for my school,
and have been trying to focus whatever education time
I have on furthering our development goals.

The network consists of some Pentium IIs and IIIs that I
salvaged from scrap and some more powerful units that
are wasting their CPU cycles and memory capacity. In fact,
I need to be able to run some of the relatively graphics-
intensive programs, Flash etc., without their slowing down
so much as they do now.

Maybe the best way is, as you recommend, set up another
server that I can afford to have messed up beyond repair
if I make too big a mistake. Even then I need to learn
about chroot. The man instructions are so hard to
understand and I haven't found any specific steps to
create a sandbox.

Other route to consider is to use VMs. That way you can just delete them
as you tire of them. Also you can try something like Gogrid.com which allows you
setup virtual servers over the internet. You can make as many or as few as you
like and only pay for the time that they run. It is very resonably priced and
you can set them up in minutes. You can make Windows, Redhat and Ruby on Rails
servers right now, but there will likely be more in the future.