Duplicate Content Penalties
People worry about duplicate content penalties in a lot of different situations. One of the more popular comes when it's done intentionally with article marketing, with how content management systems archive content, and when scrapers are stealing your content. Let's address each form in turn.
Article Marketing
If you post the same articles on several different directories, you are not going to get penalized and neither are they. What I have seen is that the more times your article gets posted around the web, the less value you see out of the incoming links from those articles. So if your article is distributed to 100 sites instead of only 10, it's not going to be 10 times as effective for you.
Think about this from Google's perspective, when a user enters a search query, do they want to list hundreds of the same articles? No, because if the user doesn't find what they are looking for in the first they want to move on to something different.
If you are using the same articles on your site as you do on others, simply putting a link back to the original article on your site will let Google know which article should appear in the index.
Your Own Site
You obviously don't want to intentionally create multiple pages, subdomains, or domains with the same content. That is against Google's Webmaster Guidelines and could cause you to be penalized. There isn't an automatic filter or penalty though. If you read Google's definition of duplicate content they specifically say it is not grounds for action unless it appears to be deceptive or used to manipulate search results.
One problem I do see on sites is when they use really short articles. If your template is really big and you have a lot of links in your menus then a short article can appear to be more than 50% the same as other content on your site. Two ways you can help to make sure you don't run into this trouble. One, is to cut down on the size of your menus and the second is to make sure you use at least 250 words of original content in all of your articles.
Scrapers
I'll define scrapers as anyone who takes the content from your site without permission. Most of the time these sites violate Google's Webmaster Guidelines since they have little to no original content of their own.
Your site is not going to get penalized for having scrapers pick up your content. Again, think about it from Google's perspective. If a competitor is outranking me could I really get them banned by just creating 100 fake blogs and scrape all of their content? That wouldn't make sense.
The problem that you may run into is Google index's the scrapers version of your content instead of your own. However, the chances of that happening on an established site is pretty rare since the scrapers are normally pretty low quality. And Google recently launched a campaign to weaken the ranking of content farming sites.