A lot of the entities we deal with in web apps are complex, multi-table structures when stored in a relational database. Think about blog articles (tables: article, author, tag, comment), user profiles (tables: user, friends, follower, interests, etc.), photos, tweets, ads, and such. The beauty of something like MongoDB is that you can store all the data as one "record" in a collection. You can also retrieve it with a simple query. These "records" are structured as something like JSON, so you can just take the data structures you're using in your app's code (particularly Python or JavaScript), and store it as-is with no conversion.

Compare this to an RDBMS-based web app, where to display a blog article for example you'd need to make multiple queries of different tables: first the "article" table to get the text, then the "author" table, then all the comments, tags, images, or whatever. The NoSQL way is much easier to code for this sort of thing. The lack of ACID doesn't matter so much because these kinds of data are usually written just once, read many times, edited rarely, so there are few opportunities for inconsistencies to creep in.