A manual action is a penalty applied by a person at Google, rather than by an algorithm, when a site breaks the search spam policies. It appears in the Manual actions report in Search Console.
For links, the usual reason is "unnatural links to your site": a pattern of bought, swapped or manipulative links. Affected pages can drop in rankings or disappear from results.
Fixing it means removing the bad links where possible, disavowing the rest and filing a reconsideration request explaining what you did.
Example
A site that bought hundreds of links from a guest post network receives an "unnatural inbound links" manual action.
Putting it into practice
Where to use it: Disavow file generator and White-hat link building checklist.