What is the relation between ROM Manager, ClockworkMod and Nandroid? Which one(s) do I need? So I understand the first thing I need to do with my phone, after rooting, is to back everything up. But I was confused about 3 of the apparently popular tools used for this.

For example, is Nandroid an application I run on a computer connected to the Android device, or on the device itself?

I understand ClockworkMod is a "custom recovery". What does that mean? Why do I need it? What does it mean to "flash" it?

Asides from browsing and installing new ROMs, how does ROM manager interact with the other two? Does it include them both, so I just need to download it from the market and be done with it?

I will appreciate any answer that can sort out the differences, with emphasis on what I, as an end-user, actually need to do, and whether there is any risk on those actions.
A "Recovery" is a special space in the internal memory of your phone, set aside by the manufacturer to perform diagnostics and OS upgrades.  In Windows terms, it's a rough equivalent of Boot Menu (usually accessed by pressing F8 during boot-up.)  It's invisible to the phone under normal operation, and is accessed in a special way (usually by turning the phone on while holding one or more buttons).

The problem is that the recoveries that comes with stock devices do not allow for installation of custom ROMs, only firmware images "signed" by the manufacturer.  In order to install custom ROMs (e.g. CyanogenMod for example) you will need to replace the stock Recovery image with a custom one, made by community developers.  On Android there are several custom recoveries, Clockworkmod (CWM) and Amon-Ra being the most known ones.  These custom Recoveries allow you to do a lot more things with your phone.  The biggest advantage is being able to flash unsigned custom ROMs.  Other benefits include various tools and utilities for further managing the phones.

One such utility is called "NANDroid".  It's a backup and restore tool that allows you to take a full image of all the phone's internal NAND memory (hence the name) and store it on your SD Card.  Should something go wrong with the phone's software, you can use the previously saved backup to restore the phone to a working state.

The maker of ClockworkMod Recovery also created an Android app called "ROM Manager".  This app allows you to easily upgrade the CWM recovery to a new version when one becomes available for your device.  It also allows you to manage NANDroid backups stored on the SD Card, as well as perform a slew of other useful advanced tasks.  These include fixing app/data permissions, re-partitioning of the SD Card right on the phone, downloading different custom ROMs and flashing them directly from the app.  A paid "Premium" version of the app has more advanced features, such as OTA update notifications for custom ROMs that support it, etc.