If you are the site owner who has been doing all SEO you will probably have a good idea about your links and keyword strategy and aware of any iffy tactics.
If you have inherited the site you need to get up to speed on what has been implemented on your site. General advice is not to overreact, take stock of the situation and set out your action plan. Do a full audit of links, keywords and SEO activity in general.
Any paid links or link schemes should be removed asap. Do this by following the same channels used to get the links in the first place, if that doesn't work or it's impossible try contact the site owner in a polite friendly manor.
If you aren't sure about the quality of the site linking to you, check their rankings, are they still ranking high for their brand and targeted keywords?
Look at your link profile, are most of your inbound links using your targeted keywords rather than your brand name, it'll look suspicious if they are? On high value sites try get the anchor text changed and add some diversity.
Did your link profile grow exponentially overnight? If the site wasn't mentioned in local/national news/events the spike might look like you bought links or did something suspicious, if this is the case start removing from lowest quality sites.
Lastly and probably pre Penguin, if you have inlinks from totally unrelated niches it's not going to look good.
If you have sprinkled your site with your exact targeted keywords (anchor and plain text) with some sort of keyword density target in mind, you might want to remove/replace with some synonyms.
Check your footer links, probably fine to have key navigational links, terms and social but avoid links to unrelated sites or low value contect in your site.
Wordpress sponsored links have been mentioned a lot. Many free themes come with a credit link to the theme developers and sometimes their partners. Removing this can be a bit of a pain as they come in the form of encrypted code snippets distributed across a number of files that all need to be removed at once else the theme will be break. There is a link in resources of wordpress plugins and themes that inject links. Tip When choosing a WP or other theme best avoid if there is any encryption on the page as it's probably doing something spammy.
Exact match domains where the keyword matches exactly the targeted keywords. I find this a bit hard to believe, perhaps downgraded a bit but possibly more correlation than casuation. It used to work well and webmasters/SEO's who bought exact match domains for traffic probably also pushed the boundaries a bit with other tactics. Anyway lets see how it goes for this very site.
Google say if you were hit by Penguin and you clean up you will be recover naturally, there is no need to submit a Reconsideration request (dealing with a site that has been manually penalised by google).
If you think you were caught unfairly you can submit your site to google for a manual review. But before you do I suggest you are 100% there has been nothing that falls outside of their guidelines that might have triggered the drop.