Search for People
On the main web page (not your cyworld popup page) you can manage your friends list and find people. Near the bottom you can see if any of your friends have a birthday or event happening soon. Click on 일촌관리.
There are a couple of things you can do here, manage your "interest" in friends, look at the address book, and find and add people to your friend list.
The "interest" thing is for friends you already have on your list. Here is where you can use a check box to control if they appear on the drop down list on your cyworld "friends" list. If you don't want them to, then just uncheck the box beside their name. This doesn't affect the other person, only your list, and it's easy to add them back again.
Finally, click on 일촌맺기 to search for and add friends to your "friends" list. (I'm using "friends" here, but the literal translation is 1 degree, like the whole 7 degrees of separation thing, these are the people who are closest to you.)
If you know the person's name and the email address they use to login, you can add them to your "friend" list here... however, I'm going to skip that for now, and go to the search for a friend page. Click 회원찾기.
Chose if you want to search by name, year of birth, and sex (the picture shows this option), or email address. Use the drop down list beside 찾는 조건 to make your choice. Fill in their name, then use the drop down lists to chose their year of birth and sex (남 - male, 여 - female). If you chose the search by email, another page will come up very similar to this one, but showing only one box to fill in, their email address. Finally, click on 검색 to search.
You should have a list of names now, and if you're lucky, there won't be too many. If you are unlucky, you might have to click on a lot of pages to find which person is actually your friend. I do this by left clicking on their name to bring up the menu, then click on 미니홈피 to bring up their page, then look at the pictures to see if this person is actually my friend. If not, close and repeat with the next name.
When you do find the person you're looking for, hover your mouse over 인맥관리, then chose 일촌맺기 to bring up the invitation page.
The invitation page is simple, just remember that the top blank to fill in is the nickname that you will see beside their name. The second blank is for the nickname that your friend will see beside your name on their "friends" list. Type in any message you want, and click 보내기 to send. This will send an email to them with your invitation.
Finally, accepting an invitation. I don't currently have any invitations that have been sent to me, so no screen shots. Basically, if someone sends you an invitation you'll get an email and the next time you login to cyworld you'll get a popup saying so-and-so invited you to be a "friends" list, and do you want to accept them. Click 예 to accept the invitation. Another popup will come up, I think it's to say do you want to have an "interest" in them (in other words, do you want that person to be on your drop-down list of friends), click 예, and finally a third popup (maybe, my memory is getting foggy here) comes up, I can't remember what this one's for, but just click 예. That should be all... next time I get an invitation I'll try to get some screen shots up here.