Friendship in Friendica can take on a great many different meanings. But let's keep it simple, you want to be friends with somebody. How do you do it?
Friendica has two different kinds of "addressbook": The directory of the Friendica server you are registered on and the global directory that collects account information across all Friendica instances.
The first thing you can do is look at the Directory. The directory is split up into two parts. If you click the directory button, you will be presented with a list of all members (who chose to be listed) on your server.
You'll also see a link to the Global Directory. If you click through to the global directory, you will be presented with a list of everybody who chose to be listed across all instances of Friendica. You will also see a "Show Community Forums" link, which will direct you to Groups, Forums and Fanpages. You connect to people, groups and forums in the same way, except groups and forums will automatically accept your introduction request, whereas a human will approve you manually.
Visit their profile. Just beneath their profile picture will be the word 'Connect' (we're assuming this is an English language profile). Click that 'Connect' button. It will take you to a 'Connect' form.
The form is going to ask you for your Identity Address. This is necessary so that this person's website can find yours.
What do you put in the box?
If your Friendica site is called "demo.friendica.com" and your username/nickname on that site is "bob", you would put in "firstname.lastname@example.org".
Notice this looks just like an email address. It was meant to be that way. It's easy for people to remember.
You could also put in the URL of your "home" page, such as "http://demo.friendica.com/profile/bob", but the email-style address is certainly easier.
When you've submitted the connection page, it will take you back to your own site where you must then login (if necessary) and verify the connection request on your site. Once you've done this, the two websites can communicate with each other to complete the process (after your new friend has approved the request).
If you already know somebody's Identity Address, you can enter it in the "connect" box on your "Contacts" page. This will take you through a similar process.
You can also use your Identity Address or other people's Identity Addresses to become friends across networks. The list of possible networks is growing all the time. If you know (for instance) "bob" on gnusocial.de (a GNU Social site) you could put email@example.com into your Contact page and become friends across networks. (Or you can put in the URL to Bob's gnusocial.de page if you wish).
You can do the same for Twitter accounts and Diaspora accounts.
In fact, you can "follow" almost anybody or any website that produces a syndication feed (RSS/Atom,etc.). If we can find an information stream and a name to attach to the contact, we'll try to connect with them.
If you have supplied your mailbox connection information on your Settings page, you can enter the email address of anybody that has sent you a message recently and have their email messages show up in your social stream. You can also reply to them from within Friendica.
People can also become friends with you from other networks. If a friend of yours has an GNU Social account, they can become friends with you by putting your Friendica Identity Address into their GNU Social subscription dialog box. A similar mechanism is available for Diaspora members, by putting your identity address into their search bar.
Note: Some versions of GNU Social software may require the full URL to your profile and may not work with the identity address.
When somebody requests friendship you will receive a notification. You will need to approve this before the friendship is complete.
Some networks allow people to send you messages without being friends and without your approval. Friendica does not allow this by default, as it would open a gateway for spam.
When you receive a friendship notification from another Friendica member, you will have the option of allowing them as a "fan" or as a "friend". If they are a fan, they can see what you have to say, including private communications that you send to them, but not vice versa. As a friend, you can both communicate with each other.
Diaspora uses a different terminology, and you are given the option of allowing them to "share with you", or being full friends.
Once you have become friends, if you find the person constantly sends you spam or worthless information, you can "Ignore" them - without breaking off the friendship or even alerting them to the fact that you aren't interested in anything they are saying. In many ways they are like a "fan" - but they don't know this. They think they are a friend.
You can also "block" a person. This completely blocks communications with that person. They may still be able to see your public posts, as can anybody in the world, but they cannot communicate with you directly.
You can also delete a friend no matter what the friendship status - which completely removes everything relating to that person from your website.