[Edited to add: Since LinkedIn discontinued their ‘Answers’ feature, this article is now 3 tips about using LinkedIn. See below.]
1. Have a good professional profile (like a CV), that is rich in keywords for search engines (e.g. in the ‘Professional Headline’ section found under ‘Edit profile’). It should also include recommendations from people who know you. You can send a ‘Request for recommendations’ (if you do, I recommend you personalise it). However, I find the best way to get recommendations is to give them, because of the ‘law of reciprocity (and because I feel more comfortable about doing it that way).
2. Ask your network for introductions to the people you would like to reach (again, only do this if you are clear who you want to reach and why). This is one of the most powerful tools of LinkedIn, but be sure you have a good reason for wanting the introduction and don’t worry if your go-between chooses not to pass it on because they are protecting their own valued contacts.
3. People may already be talking about your particular topic, and you can quickly be seen as an expert when you add value to the discussion. It’s like being at a conference, exhibition or trade fair that specialises in your area of interest online, 24/7. Search the Groups to find the conversations. Just as if you walk into a new pub, there may be people chatting about fishing in one corner, football in another, and work somewhere else, you wouldn’t walk in wearing a sandwich board and shouting ‘buy my stuff!’ Instead, watch and listen before you join in. Don’t be sales-y or you will soon be ‘jumped on’ by other members of the group.
4. Ask and answer questions in the Answers section (you’ll find it under the More link). The best answers get rated highly, so it’s a great way of demonstrating your expertise and getting noticed.
[Edited to add: As of January 2013, LinkedIn has removed the ‘Answers’ functionality. If you find ‘Q and A’s a useful way to source information and demonstrate your expertise, I recommend www.quora.com instead.]