At networking events, there's usually time to intermingle and chat with others before and after the main event, like a workshop. This is the time to swap business cards. Prepare what's often called an "elevator speech" - imagine you are in a lift with someone, you've got 30 seconds before one of you gets out - how would you sell yourself in that time?
A good example is "Hi, I'm Louise. I run PlainText, which provides freelance copywriting and editorial services to small businesses in Chester and the UK. If you'd like to increase your sales by improving the quality of your brochure copy, I'd be pleased to help you out."
Put aside some time to practise your elevator speech beforehand - work out different ways of introducing yourself and honing your message, then try it out on a friend who's not too involved with your business to get feedback from them.
When you talk to people, ask them what they do, listen to what they say and see how you can offer to help them (you may not be able to depending what they do, but all contacts are useful). Don't blaze in selling yourself really hard as it's off-putting. Keep your chat short and sweet and then move on to someone else.
After the meeting, you can go through the business cards you collected and follow up any interesting people with an email or phone call to develop further contact.
What to wear? It's an evening do, so for men, a suit or smart casual is probably best. Avoid jeans and t-shirts even if it's 30C outside. You want to look professional. For women smart casual is best. I work from home and sit around in combats and trainers most days but if I'm networking I'll wear heels, smart trousers, a nice top and a funky jacket, and I'll put a bit of make-up on (I don't bother most days!).
If you ever go to a breakfast networking meeting, it's suits all the way.
If there's a workshop, you'll probably be expected to participate. Follow the workshop leader's instructions when you break up into small groups and take your cue from the old-timers there. Workshopping is another good way to network as you'll see how you get on with the others in your group.
Sometimes networking meetings have presentations instead of a workshop.
Also, there's often a formal round of introductions before the main event - this is another opportunity to use your elevator speech to promote yourself to the whole room.