Conflict of interest is one of Wikipedia's least understood and yet most cited behavioral guidelines. In a nutshell it strongly discourages promoting interests an editor is related to unless it is certain that the interests of Wikipedia remain paramount.
Conflicted editors can be an extremely valuable and largely untapped resource as they are often the most equipped and motivated to report on a given subject. Policy permits conflicted editors to edit articles provided they comply with other policies. Unfortunately some such editors fail to write from a neutral point of view, introduce original research or unverifiable claims, link to third-party websites or breach other policies and guidelines such as those dealing with autobiographies.
The word interest has two distinct meanings in this context which can be a source of great confusion:
While both meanings usually apply at the same time (for example an editor writing about their own company or family member), this need not always be the case. Just because an editor has an interest in a topic (regardless how strong) does not imply that they have a conflict of interest. Conversely an editor may have a conflict of interest with no interest at all (for example an editor writing about an ex-employer or competitor).
Where there is no obvious off-wiki financial, professional, or marketing benefit to the editor or someone close to them (examples) it is usually better to avoid introducing conflict of interest and focus instead on the actual violations of policy (indeed this can be good advice even when there is a conflict of interest). Many editors consider such accusations to be a personal attack and may respond aggressively where they may have otherwise remained calm.
Conflicted editors who do choose to contribute should declare their interests, both on their user pages and on the talk page of any article they edit, be sure to write in a neutral tone, cite reliable, third-party published sources, and beware of unintentional bias. If challenged on your conflict of interest, remain cool, and be willing to edit collaboratively. Using the {{edit COI}} tag on the talk page or asking for input at the conflict of interest noticeboard can resolve disputes.