While exploring you will see round icons indicating job boards with Foundry quests, also "well-informed local contacts" have such icon, they are offering "rumors" ie local foundry quests.
The Foundry is a tool used to create your own quests or even quest chains, whole campaigns even. You can create your unique story in Neverwinter world and allow other to complete it. In the cities icon indicating foundry quests will lead to job board which will open Landing Page on "Catalog" tab where all foundry quests are listed.
Sometime same icon leads to "well-informed local contact" offering you featured and local foundry quests.
While viewing foundry quests note yellow and red star ratings, those are both player ratings, but quests with red stars are featured in this week.
Some daily quests require you to complete eligible foundry quest, eligible quests have green "This quest is also eligible for the 'Daily Foundry' quest rewards." above quest details.
Catalog tab of Landing Page has several important elements. On the right side you can see details of chosen quest:
Quest overview with author's notes, information on the creator, average time to complete, links to other tasks by this author, button allowing you to subscribe this quest, and of course the Accept button the bottom. In "Map" tab you can check location preview. In "Reviews" tab you can check what other players say about this quest.
On the left side you will find more tabs: Best, Featured, Subscribed, New and For Review, you can filter the results of each tab specifying author, minimum duration, name or short code, language. After accepting foundry quest it's completion does not differ from any other quest.