Interests are a way of expressing what you like and what you are interested in. They show up in your userinfo page (http://www.livejournal.com/userinfo.bml). You can search them to find other users and communities who are interested in a certain subject.
There are several ways to add or remove interests. The most simple way is to enter them in the "Interests" section of the Edit Info page: http://www.livejournal.com/editinfo.bml. You should enter them in seperated by commas, for example: cookies, chocolate, music, swimming. Note that there is a limit of 50 characters or four words with single spaces between them. If you try to add a longer interest, it will not display on your userinfo page. To remove an interest, simply delete the text. Also note that you cannot add more than 150 interests.
You can also add them through enmasse mode. You can do this by going to someone's userinfo page and clicking [Modify Yours], which appears at the end of someone's interests. On your own page, this will appear as [Remove Some]. Once on this page, you can check checkboxes next to interests you'd like to add, and uncheck checkboxes next to interests you'd like to remove.
Another way to reach this page is from the main interests page, located here: http://www.livejournal.com/interests.bml. You can enter someone's username in the box labeled "Modify your interests based on those of:" to add some of their interests. Note that some users will be offended if you make all of your interests the same as theirs. Interests are a way of finding people who are similar to you, and by adding things you are not really interested in, you can pollute the pool of interests.
If an interest you list is listed by other people, it will appear as a link in your userinfo page. Clicking this link will take you to a list of other people who are also interested in this topic. Because of the strain it would put on the database to load all the usernames who list a popular interest, this search is limited to the first 500 users and the first 500 communities to list this as an interest. If no one else lists an interest you have, it will not appear as a link, but as plain text.
Another way to search for interests is via the main interests page: http://www.livejournal.com/interests.bml. By typing an interest in the box labelled "Find people and communities interested in:" you can find people who are interested in the selected topic. From that page, you can click the words "click here" to add the interest to your list of interests.
Finally, you can find the most popular interests here: http://www.livejournal.com/interests.bml?view=popular.