Over the past several months Google has been rolling out algorithm updates in order to fight off spam and make the search results better and more reliable than its competitors; being Yahoo and Bing. The penguin update was first released in April of 2012 and has been updated this last May 2013. This algorithm change was targeting websites that violate Google Webmaster Guidelines by partaking in cloaking, keyword stuffing, and other blackhat SEO methods. Sites participating in such schemes to manipulate the search results were penalized and lost valuable search engine rankings.
In order for a website to regain its old rankings in Google it must clean up its backlink profile. There are many free and paid tools that will show you all your backlinks and help you analyze which are good and which should be deleted. A good tool that shows in depth analysis of each backlink in an easy to read manner is Ahrefs. This is a paid tool, but has a free area that will show you a few of your websites backlinks. This is a good start, but you really need to look at all your backlinks using a variety of tools. Always check Google Webmaster Tools to see what links Google sees pointed to your site. Analyze the links and see if any seem like low quality SPAM links and if so try to remove them.
Removing links can be a difficult process. Often times the sites is dead, meaning the owner does not care about it and nobody manages over it. These can be a problem as you have to wait for the domain to expire to get the link permanently removed. If the site is active you can contact the webmaster using the contact form or about us page. If it is not listed there you can do a WHOIS check to see who registered the domain and contact them this way. Contact the webmaster and politely ask them to remove the link. Sometimes they will be nice enough to do it and other times they may want compensation. I would not spend more than $20 to have a link removed as there is another option using the Google Disavow Tool. Using this you can enter a list of website URLs that are negatively impacting your websites rankings to let Google know to ignore these links. This is a new tool and has some controversy, but it can be very helpful if your website was hit by the latest algorithm update.
In essence it is best to practice good, ethical, white hat SEO techniques so you do not have to worry about Google and their algorithm changes impacting your websites search rankings. Write good, quality content that is shareable and helpful to your visitors. Answer questions that clients commonly ask as these will be highly searched for terms. This is the best way to do SEO and will ensure you never get hit with a Google penalty.