Open meeting - non-alcoholics can come (e.g. hand holders, but also interested members of the press, GPs, police etc) - though they may not share or contribute to the collection at the end (because we are self-supporting.)
Closed meeting - only for alcoholics (in the broadest sense of people who think they have a problem with drink, even if they have not yet decided to abstain and even if they don't like the A word.)
Quite separately, there are also different types of meetings (which, for obvious reasons, all tend to be "closed".)
The vast majority of meetings are not of any particular type and involve an alcoholic of six months sobriety or more telling his or her story (what it was like, what happened and what it's like now) for about half an hour, then other people present "share" with the speaker, talking about how his or her story is like theirs. This is normally through "raised voice" sharing; if you want to speak, you wait till the person before you finishes, introduce yourself and say what you have to say. Etiquette is to talk about yourself, not others; to shut up after a maximum of 5 minutes (or less if that's the rule in the meeting) and not to share if you've had a drink today. (I have to confess I have borken all three of these rules in my time ...)
Typically, this sort of meeting will start with a brief reading from AA literature of the speaker's choice.
A Step meeting is identical, except that the reading will be one of the 12 Steps, the speaker will try to base his or her share around that step, and the people sharing back from the floor will also try to relate their sharing to that Step.
Big Book - same again, except the reading will be a passage from the Big Book (often read in order, week by week, or according to a pre-determined list) and, again, sharing themed around the reading.
You also sometimes find "Living Sober", "Reflections" and "As Bill Sees It" meetings (based, surprisingly enough, around the books "Living Sober", "Reflections" and "As Bill Sees It".)
In other words, the difference is that the speaker is not completely free to share about what he or she wants to.
That said, whatever type of meeting it is, if I know there's a newcomer in the room, I will tend to include more reference to my drinking and to how I came to AA in my sharing, and I think this is normal - we want you to identify with us, in other words, to realise that you are not alone and that (in my case) seven years off the drink nevertheless started with a day exactly like the one the newcomer's having today.
Incidentally, in big cities you sometimes also find meetings describing themselves as "Newcomers'" meetings. In fact, this is a very flexible definition, but you will find that more of the people present will be in, say, the first 18 months of sobriety. Naturally, this also means that they tend to be younger.
My advice to you would be to go to bog-standard meetings initially (as opposed to newcomers' meetings) - not only will you get the benefit of a breadth of experience not always available in newcomers' meetings, but it will help you realise that all kinds of people can be alkies and that some of the most respectable faces in your local community used to be piss artists and turned it around.
I would also warn you that, while Big Book and Step meetings can be excellent, they may initially seem a bit theoretical and removed from your immediate problems. Be sure to tell the greeter and/or the secretary on the quiet that it is your first meeting and that you have not been "dry" long and they will help mitigate this effect. But don't feel you can't or shouldn't go - such meetings can be a huge help (and sometimes they go back to Step 1 when they know a newcomer is present.)
Happy to go into more detail if that would help. Where are you based?