DO invite me to stuff.
DON'T think my turning down an invitation means I don't like you.
That pretty much covers it, really. Everything else is my issue, and other people shouldn't have to think about it. But if you want to think about it, here is some background information, with additional suggestions for application.
Being an introvert means that spending time with people takes effort. That doesn't mean I don't like it, but it does mean I can't do it all the time. Alone time is like sleep. It's probably not the most fun, exciting thing I could be doing at any given moment, but I need it regularly or I won't function well. Also like sleep, I can be pretty flexible about whether I get it right now, but I will need it some time that day.
Being around people, on the other hand, is like exercise. It's fun, and I need it on a regular basis to function properly, but it tires me out. Also like exercise, if I have been getting very little social time over the past month or so, it will be much harder to interact than if I've been getting a nice regular amount of social interaction. Too much is bad, and takes a while to recover from, but too little is also harmful in the long run.
I am good at regulating the amount of social time I get in a given time period to stay within acceptable bounds, as long as I can plan it in advance. Spontaneous gatherings are tricky; so are last-minute cancellations. (In general, I have trouble with sudden changes of plan. I would like to be more comfortable doing things spontaneously when I do have the social energy to do them; this will require some practice.)
I can handle significantly more social interaction than I think to initiate. Particularly, I tend not to issue a lot of invitations for one-on-one time (I get all nervous about being forward or pushy, especially with people I don't know all that well but would like to know better), but I'm likely to accept them or, if I'm busy, suggest an alternative activity or time. Likewise, replying to someone's e-mail is much easier than writing them a brand new one. If you particularly want a reply to an e-mail, include a question mark in it somewhere; I'm pretty good about replying to questions reasonably quickly, even if I'm having trouble thinking of other things to write about.
Interacting with new people is substantially more difficult than interacting with people I already know. I think of this as sort of an investment; it takes a certain amount of social energy to start a social interaction, but some time after the interaction (if it goes well) I will get social energy back, possibly more than I started with. With new people, the initial cost of the interaction is higher and I don't know whether I'll get that energy back. It's a riskier investment, so I don't make it unless I've got a reasonable amount of social energy banked up. In other words, I like meeting new people one or two at a time, in amongst other people I already know, rather than in big clumps.
Things that are helpful:
1. Making plans in advance. Not necessarily terribly far in advance; I don't need weeks of notice. But if I am surprised by an activity beginning, I will need to spend the next ten minutes or so resetting my expectations instead of enjoying whatever it is. (The resetting is not all that annoying, but other people can tell that I'm not having fun during it, and tend to conclude that I don't like the activity or the people involved. Understandable, but incorrect.)
2. Predicting the size of an event. Sometimes, particularly if I'm heavily scheduled, I may have the energy to deal with a few people but not lots of them. It's helpful for planning purposes if I know whether you're inviting me to dinner or a movie or whatever it is with just you, with you and a person or two I already know, with you and a person or two I don't know, with lots of people a substantial fraction of whom I know, or with lots of people I mostly don't know.
3. Providing quiet space. At very large or noisy events (or if I'm otherwise stressed), I may get overwhelmed and need to flee. If there's a quiet area handy, a brief retreat is often sufficient to restore my equilibrium. If there's not, I usually try to stick it out for the sake of whoever I'm there with/for, but it's not good for me. I tend not to go to things that seem likely to be stressful unless some particular person really wants me to; knowing that there's a retreat area available would make it a lot more likely for me to attend crowded events. Having someone to run interference can increase the effectiveness of quiet space; it's frustrating to have people coming up and checking to see if I'm okay when I would be fine if only people would stop asking me about it.
4. Giving me time to lurk. Especially for things like classes and workshops, where there are a lot of strangers, I like to watch what's going on for a while before I join in. Being heartily welcomed and included in the group activity right away is not good, unless I already know who most of the people are and what it is they're doing. A quiet greeting from one person, with some general orientation or indication of who to go to with questions, is much more comfortable in a new situation. A good clear briefing ahead of time will also work.