When a user searches they very often use a phrase. And some engines actually encourage this as this is human behaviour. Stop words are prepositions, joining words or just fluff (wiki definition: Stop words). These words are very common but don’t add any value to the users search.
To-date search engines that focus upon retrieving search results on the users keywords found in queries have often ignored stop words.
“Stop words” can be ‘a’, ‘on’, ‘the’, ‘in’. And sometimes stop words can be joined and that is called a “stop phrase”
A practical example in the travel space is “hotels in new york”, the ‘in’ is discounted. The exception is when a preposition is part of the exact phrase match e.g. “alice in wonderland”, the ‘in’ is needed here. The major players are reported to actively working on ways to understand if they add value! Good luck to them.
Why are stop words important? Well everything you do regarding keywords is about linguistics. If you today search for a phrase with and without stop terms depending on the query you may see different results. If you are working in PPC and you are extensively using exact match, you need to set up differently.
Until the search engines fully understand human language (they are patent heavy in this area so I guess they are on it) then how you optimise on and off page using stop words still makes another small difference.