Introverts like me have an extra layer of difficulty with this sort of thing, as does anyone with a disability or mental illness. Friendships and support networks can be really hard to build and maintain if social contact itself makes you uncomfortable or nervous, and if you are periodically (or perpetually) needy. Or if you are the sort of person who is only comfortable having a couple of close friendships.
So let's consider how introverts can have a healthy, enjoyable social life.
First, understand where you fall on the introvert-extrovert scale. If you sometimes function in introvert mode and other times in extrovert mode, try to identify the circumstances for each and whether there are any switches you can throw on purpose. Next, understand that both introversion and extroversion can be healthy. Your social orientation is only a problem if it makes you unhappy and/or prevents you from doing things you want or need to do, in which case you can work on shifting it. If you're comfortable with it, leave it as it is and work on finding other folks who will accept you as you are.
Second, consider your wants and needs in the social field. What kind of friends would you like to have? How many? How often would you like to see them? What kind of things would you like to do together? Is your family life the way you want it? If not, what's missing? Can you fill in the gaps by boosting some existing relationships, or do you need to find/create new ones? How many social activities would you like in a month? How long can you be around people before it stops being fun and starts to bother you? Also check to see whether you have too much going on in this part of your life. If you feel overloaded, what can you reduce? Here's a list of graceful ways to say "no" so as to avoid overloading yourself further.
Third, consider your talents, skills, and virtues. What are you good at doing? What do you enjoy doing? Where are some areas that your endurance is higher than other people's is? What do you enjoy doing, or at least not mind doing, that other people dislike? What do you take pride in? When X happens, people turn to you -- fill in that blank. Also think about things you can't do yet, but would like to learn; you can keep an eye out for opportunities.
Don't Use a Screwdriver to Pound Nails
Modern American culture tends to favor the extrovert mode. That doesn't make it better than the introvert mode, just more popular here and now. Trying to make an introvert act like an extrovert is like using a screwdriver to pound nails: it might eventually get the job done, but not very efficiently and you're likely to smack your thumb or scratch the wood in the process. The trick to building a happy social life for an introvert is to find ways to play to your strengths, not your weaknesses. Understand that you're a jolly fine screwdriver, and there are plenty of screws out there. Not everything is a nail in need of a hammer, so don't try to be a hammer. Be yourself.
Show, don't tell. Many social events are intended to "tell" people that they are loved and appreciated. There are lots of ways you can "show" people instead, which may make them less insistent about dragging you into a busy event. Many introverts are good at "fading into the woodwork," which makes it easy to observe people and learn what they like and care about. You can then make their favorite food, play music they enjoy, surprise them with the perfect gift, etc. This is a great way to contribute something to a relationship or community, to balance whatever you get out of it.
Do solitary tasks. Some careers necessitate spending lots of time by yourself; these are better done by introverts than by extroverts who would be miserable. Some events require setup tasks that would bore an extrovert -- somebody has to make the cake, fill the balloons, hang the streamers, plan the menu, etc. This is a way to take part in an occasion that's important to you, without getting swamped by a crowd. In a well-balanced group, someone at the event will pass the word as to who did which of the cool setup bits, then pipe the feedback to you later. Another example is community theatre -- not everyone is an actor, and the people who make the costumes and the sets lend a great deal of magic to the final performance. Some introverts become quietly famous "behind the scenes" for making things happen in their community, and are valued accordingly.
Be a good listener. This is not a common skill in modern America, but it's an extremely valuable one. Some introverts are perfectly okay with people as long as they don't have to do the talking themselves; these often make great listeners. A good listener is a good friend. Even if people make you nervous, you can use listening skills to lighten the load by getting them to talk about themselves so don't have to be there and talk.
Practice companionable silences. Extroverts are great if you want to talk, but sometimes you think, "ZOMG if I get asked about this one more time I'm gonna scream." Then it's time to hang out with your introvert friends who will not pester you. Also, some activities lend themselves well to companionable silences, such as quiet walks through the woods or along a beach. Some people enjoy doing crafts together and don't feel compelled to fill the air with words the whole time. These are good choices for introverts who are friends with each other.
Explore quiet groups. Not all activities are boisterous ones. An introvert who feels over-stimulated by a party might feel comfortable in a class of yoga, meditation, T'ai Chi. A birdwatching group might begin with a few minutes of greeting and discussion followed by several hours of relatively quiet hiking because you're trying not to scare all the birds away. Also, if you don't want individual friends bugging you constantly, it can be very effective to socialize at defined times and places on a weekly or monthly schedule.
Communicate clearly. This can be challenging for introverts, but it usually pays off, because misunderstandings so often cause tension. Most of the time, you'll get better results from telling someone "I can be there for X hours" than just showing up and then trying to get loose early. If possible, you should let your friends know that you are indeed an introvert, so they'll understand when you say, "It's been fun, but I'm getting all peopled out now so I need to go home." They're less likely to get upset if they understand that they haven't bored or offended you, it's just a matter of your internal limits. This also applies to limiting the number or size of your social engagements, as well as the length.
Offer alternatives. People often want to include folks they care about, but they may not understand your strengths and weaknesses. If they suggest something that would be unworkable for you, say something like, "I can't do X, but I could do Y instead." Learn what kind of things you do that other folks find helpful, so that you've got a ready list of alternatives.
Maintain long-distance relationships. If you're not comfortable having friends underfoot all the time, but you want to maintain some connections, try doing it at arm's reach. The Internet is your friend, and your path to making as many friends as you want, who will not show up on your doorstep because they live halfway across the country. You can share to the degree that feels right to you; most social networks have tools especially designed to help you control who can access which information. Similarly, keep touch with distant relatives by exchanging cards, gifts, and personal news rather than frequent visits.
What are some other ways you have found, either as an introvert yourself or an extrovert who knows introverts, to use this mode of interaction for fun healthy socializing?