Introduction
Any Ubuntu Cd can also be used as a LiveCd as well as an installer. Windows people might be more familiar with the term 'boot cd' or 'bootable cd' but a "LiveCd session" is typically much more useful because it gives a normal desktop environment and all the normal programs - it should find the machine's internet connection and allow you to use Firefox from the top panel/taskbar to surf the internet.
Most other versions of linux (distros), not just Ubuntu, have this functionality on their installer Cd too. With most other distros you usually have to click 2 or 3 buttons to configure the network connection so it tends to be fairly easy but not as easy as Ubuntu 
While a LiveCd session is a good way to demo Ubuntu to see how it works "without making changes to this machine" it is slower than a full proper install running from a hard-drive. Cd/Dvd-drives tend to be a lot slower at read/writes than hard-drives. LiveCd sessions can also be tricky to keep data and settings between sessions but you can always use a 'persistent image', perhaps on a usb-stick - see later.
Reasons for Using a LiveCd Session
LiveCd sessions are good to use for;
- a quick 'demo' on a machine before installing/upgrading Ubuntu
- to check the hardware works as expected
to see if you like the look & feel of the distro
to prepare yourself & your unique hardware if required
- various repair tasks such as
repairing grub perhaps after (re)installing Windows
fixing Windows problems on a machine that doesn't have a dual-boot
anti-virus issues
resizing partitions to give Ubuntu more (or less) room
adding a new partition(s) to your hard-drive for other distros
- preparing a machine for installing Ubuntu onto if you are shy of using default or automatic settings
- 'showing off' Ubuntu to people on their own machine
- a safer and faster way of surfing the internet or even other uses of a machine that isn't your own. This is particularly good if you have a usb stick or something to save your data and settings on (see 'persistent image')
Other Trial Versions of Ubuntu
For another way of 'trying out' or having a longer demo of Ubuntu it might be worth considering installing Ubuntu inside Windows using the Wubi installer https://help.ubuntu.com/community/Wubi or https://wiki.ubuntu.com/WubiGuide
However, while the Wubi works very well on many machines and some people have even managed to keep it working for a few years it is really not meant for long-term use. I would tend to use LiveCd sessions with a persistent image rather than the Wubi although the Wubi is very much faster. Since it depends on Windows systems to work properly the Wubi is a bit like building a house on shifting sands. Also un-installing the Wubi and removing all traces of it is tough. I find a proper full dual-boot is much easier to set-up and also to remove. This is also much more robust than the Wubi as a proper dual-boot uses linux file-systems such as ext3 or ext4 and the linux boot-loaders such as Grub or Lilo https://help.ubuntu.com/community/WindowsDualBoot
Most linux users have a dual-boot, usually with Windows in the mix somewhere. Multi-boot systems are usually still called dual-boot even if they have far more than 2 OS's to choose from. Installing Windows is a very tough challenge so it's better to avoid wiping it even if you find you have not used it in years. An experienced linux user or someone with considerable technical knowledge about Windows might find it better to try the Wubi in some certain circumstances where a proper decent install of Ubuntu is infeasible or inadvisable.
How-To LiveCD
Preparation
You need an Ubuntu Cd although it's also possible to get a "LiveUsb session" from a Usb stick and this is often also referred to as a "LiveCd session" because it does offer much the same functionality, perhaps a little more. To get an Ubuntu Cd or 2 either use the Shipit service which can can 12 weeks (but it's always nice to have a free official Cd or a few for giving away to other people (selling them is not really allowed although you are allowed to recoup costs incurred and for technical support if you help them install Ubuntu or other distro, read through the GPL licence for further info)) https://shipit.ubuntu.com/ and/or download it from http://www.ubuntu.com/getubuntu/download to make your own cd of it much faster. I tend to find the cheapest "write once" Cds that can only usually be bought in packets of 10 or more are much better for this than more expensive cds. Dvd's tend to be really rubbish for this. Low 'burn' speeds are better. Here's a guide to help if that is difficult, as it often is first time https://help.ubuntu.com/community/BurningIsoHowto
Another guide to help is Getting Ubuntu if any of this is being awkward for you. Don't worry. These things are seldom easy first time but once you have done this once or twice then it's really very easy most of the time.
The How-To
So, put the Ubuntu Cd into the cd/dvd-drive and reboot the computer. This should hopefully get you to a menu with "Try Ubuntu without any change to your computer" as the top, default, option. If you don't get this menu then this guide should be able to help https://help.ubuntu.com/community/BootFromCD
Choosing the "Try Ubuntu without any change to your computer" should get you to a working desktop which we call a "LiveCd session". Exploring the Boot Options should help if you have troubles getting the working desktop but if it is still not behaving then try asking for help https://answers.launchpad.net/ubuntu/+addquestion or https://help.ubuntu.com/community/Signpost/Questions#help
Note that Server Edition works quite differently in that it doesn't have a gui desktop so it's LiveCd gets yu to a working command-line rather than a working desktop - some other distros also run from the command-line rather than having a desktop. Anyway, from most LiveCd sessions Firefox on the top panel/taskbar should easily surf the internet and of course almost all the other normal programs are in the menu's along with one or two more.
After you have finished, shut the computer down and remove the Cd. At this point anything you saved to the desktop or Documents folders and such will vanish - only things you saved into folders on the hard-drive will remain.
One reason why a LiveCd session is so useful on machines that don't belong to you is that none of your personal data nor emails nor traces of financial transactions remains on the machine. There will only be traces of on-line activity at the isp, router, server or other monitoring system level.
Installing Programs or Other Packages
Note that while the LiveCd doesn't give much space it is possible to install programs to a LiveCd session although these will be forgotten as soon as you switch the machine off. Still it can be handy for a few things such as installing and using an antivirus on a Windows hard-drive without booting into the hard-drive, or for installing data-recovery tools again without touching the hard-drive
Persistent Image, keeping your setting
If you want to save your LiveCd session to a Usb stick, check LiveCDPersistence
Making a Customised LiveCd
If you want to make custom Ubuntu-based Live Cd, you should read the LiveCD Customization Howto. Without even starting from an existing Ubuntu Cd you can make a customised live system, see LiveCD Customization From Scratch.
Troubles with a LiveCd
Logging in
Sometimes a LiveCd might ask you for a user-name or password. Just leaving these entry lines blank and pressing enter (or allowing it to time-out) should get you past this step without troubles.
== Security and Updating ==
While linux systems are very much more secure than Windows it is worth noting that LiveCd sessions are not meant for long-term use nor for long sessions lasting for days. https://help.ubuntu.com/community/Antivirus
It is not a good idea to update a normal LiveCd session because it will quickly run out of space. This means that known issues, exploits and vulnerabilities that are usually quickly patched in linux will be left open on the LiveCd. Also running a LiveCd session still leaves you vulnerable to phishing and also may allow a variety of attacks against your network or allow your machine to be used as a launch-place for a variety of attacks on other systems either on your network or externally. There are other security threats i don't fully understand although some comments near the end of http://distrowatch.com/weekly.php?issue=20090803&mode=67 and at the beginning of http://distrowatch.com/weekly.php?issue=20090810&mode=67 might explain some things. Mostly this will not be a problem because the OS and programs cannot be subverted because the Cd cannot be written too, a LiveUsb or use of a persistent image removes this protection. Anyway, as you have seen from the Antivirus link it is very rare for linux to suffer from security issues due to inherent design, there is no anti-malware industry depending on security flaws in order to generate income streams.
While such problems might be incredibly rare and non-existent except in theoretical terms (especially in the Ubuntu LiveCd as opposed to most other distros LiveCd) it is worth being aware that there are issues to consider.
A persistent image removes a lot of the advantages of running a LiveCd session but does add back a lot more, especially in giving more space to allow for all the normal updates to be applied.
Setting a Password
You can set a password during a LiveCd session by opening a terminal, and typing in:
$ sudo passwd ubuntu
Text-Mode Installation
If your normal installation fails without giving an error message, or if you want to install on a very limited system, you can use the text-based installer instead.