Empire is a Client/Server game. Somewhere, someplace, sometime, a deity decides to run a game. That deity loads the server portion of the software on his machine. Once the deity has the game parameters decided upon, the server built and up and running, the deity will usually announce that they are accepting applications to play in the game. These announcements are usually done on the Usenet newsgroup 'rec.games.empire'. These are called 'long term games'.
Game announcements may also appear on the games page. In addition to these 'long term games', there are also usually a few games which are constantly running and reset, called 'blitzes'. The main difference between 'long term games' and 'blitzes' is that 'long term games' usually only have one or two updates per day and can last up to a few months (even longer!) in duration, whereas 'blitzes' usually have many updates per day and last anywhere from a few hours to a week. Games are usually listed along with their parameters (such as when the updates occur, how many updates, how to sign up, and other useful things to know about the game - you will (hopefully) learn what these useful things are as you go.)
Once you have found a game you would like to play in, the next thing you need to do is to get a client. It is highly recommended that you first get a client, then connect to one of the 'blitzes' listed on the games page to make sure your client is working correctly. The two most important things you will need to know about a game to connect are the 'HOST' and the 'PORT'. Each client is usually configured slightly differently, so check the client documentation on how to enter this information.
After you have connected to a game, you will be prompted for a country name and password. If you are trying the blitzes, try a number between 1 and 10 as the country name (most blitzes reset with just country numbers so that anyone can jump in and start playing, and most usually have at least 10 countries. Some have more, so try other numbers. And, most have a 'visitor' country with 'visitor' as a password so you can peek in on what is going on, you just can't actively play.) Use the same number you used for the country name as the password. If you have signed up for a long term game, you will want to use the country name and password you signed up with.
If you can connect and get the 'Country Name?' prompt, but cannot get in, don't fret, your client is working properly. You will need to contact the deity of the game you signed up for to find out the proper country name and password, or in the case of a blitz, it may just be full and you have to wait for the next time it resets. But, you have connected, and your client is functioning properly. Various clients have different places where you enter the country name and password, but you should at least get a useful message if you can connect but cannot get into a country.
Once you have managed to log into a country, you are ready to start playing. One thing you may want to review is the info pages (these are also available in the game by using the 'info' command.)
Note that as a beginning player you will often be referred to as "fodder". This is a term used liberally to describe players of lesser experience. Fret not. Ask questions on how to play the game in the rec.games.empire newsgroup. There will be plenty of people to help you.
And, lastly, good luck (you're gonna need it!) Muhahahahaha...