What is "rooting?"
Rooting your Android phone is, in a nut shell, a process in which you gain access to abilities of your phone that were unavailable before root, also known as "root access." Rooting makes you a "super user" so to say, in that you can install custom Android OS ROMs, which can be helpful when your phone's manufacturer/provider decides that your phone will no longer receive firmware/software updates, but you still want to receive the latest versions of Android on your phone.
Don't know what ROMs are? Your Android phone is running a ROM right now. It is the Android OS version that was installed on your phone by the manufacturer and likely updated by your carrier. A custom ROM is a ROM that has been altered by a third-party developer/hacker/whatever you want to call them to work on specific phones.
Benefits and dangers of rooting:
There are many phones out there that have custom ROMs made by the Android community, which bring with them new and useful abilities that would not be available if the phone only ran the official version supplied to the phone. There is not a ROM for Android 2.1 or Android 2.2 for every phone out there, so if that is the main reason for rooting, take a look before hand and make sure there is a ROM out there for you.
In addition, ROMs are usually not as stable and complete as the version that originally come on your phone. There may be some important features that are missing from these custom ROMs which make it worthless for some users.
Rooting can also give you the ability to remove bloatware/unwanted applications that came pre-installed on your phone from your carrier or phone manufacturer, overclock your phone's processor for faster performance, and take screenshots of your phone, as well as many other valuable abilities.
Rooting can be dangerous for the phone, however. Increasingly, there are "one click" root methods that can be found on the Android Market or websites that specialize in Android and the ROM community. These are very easy methods for rooting your phone, which usually involve only installing the app, opening it, and clicking a button that would say something like "root me." I do not know the success rate from those rooting methods, so be sure to check the feedback from those methods beforehand to get a rough idea whether it will work or not.
It wasn't always like that, as those of you who have been around for a while can testify to that. There are still many phones (in fact, I would venture to say most phones) that have to be rooted the old fashioned way. The old fashioned way usually required a decent knowledge of programming, a willingness to follow the directions exactly, many steps, and could often result in a ruined phone. The desired end result is typically the same though, it just requires a significantly increased amount of work to achieve root.
When a phone is ruined during the rooting process, it is usually as useless as a brick or paperweight, thus the term "bricking your phone."
As far as I know, not every phone has been rooted, and even if a phone has been rooted, the rooting method may not have been released, or may be fairly complicated to accomplish.
Rooting can get in the way of an official update to your handset. I have read of instances where someone rooted their phone, and when an official update came around to unrooted phones, those people with rooted phones had to wait for someone to get a fix for them, or unroot their phones so that the official update can be installed.
To root or not to root?
So, should you consider rooting? It is a question that only you can answer. Ask yourself: Are these abilities mentioned above important enough for me, that I am willing to begin this journey, reaping the benefits of the higher level of control over my phone? Am I willing to accept that my phone may be ruined in the process if I make a mistake? Does knowing that rooting my phone may present some problems in the future take precedence over not having this control?
If you are new to the Android community (which you likely will be, since you are reading this) and just bought your phone only a couple weeks/months ago, I personally would recommend waiting a while to root. In addition, if you don't have extra money to spend on a new phone, (in case you do brick your phone) then I would advise against rooting, until you feel very confident that you really want it and can accomplish it safely.
If the answer is "yes, I still want to root my phone", then let me direct you here, where you can start your journey on the road to rooting.
I take no responsibility for any problems, issues, or any outcome resulting from anyone who has an undesirable outcome after reading this post. Know that you are the one initiating this process. This is merely an attempt to inform Android users of what rooting is and the possible outcomes from rooting.