You received a great reply from Alex.
We have lots of problems with content theft and go after much of it. Our response depends on WHO has grabbed the content and republished it. If our content is used on a dirtbag spammer site we go straight to the DMCA form that Alex linked to. Google responds to them, often the same day, and almost always within three or four days. They remove the infringing content from Google search. We have done this hundreds of times. (If you have many to file, you can use the back button of the DMCA form, edit what is different and send it in again).
If the content is on a large blogging site like WordPress or on a large forum, Google often tells us to go there and have the page taken down. These sites usually respond promptly.
Before you go cranking out DMCA at scale, if is a good idea to consult with a copyright attorney to learn about fair use or similar concepts in your country. There are many ways that someone can use your content without breaking copyright law. If you have someone taken down where fair use applies or where your claim to copyright can not be documented, they you could be on the receiving end of civil legal action.
Sometimes, more frequently than you think, we find our content on sites that should damn well know better than take our content. These have included law firms of all sizes, police departments, district attorneys, government regulatory agencies, government law enforcement, or simply the website of an honest and respectable company. In these cases we don't go to DMCA. These folks need education. So our attorney gets in touch with firm but friendly greetings. The culprit is often someone at their office who works on the website and took some shortcuts in content development. Also, their SEO or web developer grabbed our content, added it to the website and charged them for the content. These cases of content theft piss you off, but watching what happens can be entertaining.
ADDED: I forgot to address your main question. Can content theft and republication hurt your rankings? YES IT CAN. Some of the sites who steal content are powerful and can outrank you. In addition, you can lose long-tail traffic because your words on their page combine with other words on their page and outrank you in the long tail.
If a lot of websites steal the same piece of content, Google can have trouble deciding who owns it - even if you have a reasonably powerful site and the content has been on your website for a decade. Google says that they are "good" at identifying the owner, but I don't agree. We had important pieces of content descend into supplemental search results after a lot of other sites had stolen it. It only emerged after we spent a lot of energy on DMCAs and direct legal notices.