Extravereted Intuition - Ne
Ne is an extraverted Perceiving function, meaning that it gathers information from the external world. Ne shares the spontineity of Se, except instead of seeing "what is," it perceives "what could be." Instead of working on objects, it works on ideas. Ne brainstorms, envisioning all the possibilities, creating new solutions to old problems, inventing scenarios while doing causal things (like walking down the street), perceives meaningful connections that aren't obviously related, and generally seeks to find all the possible ways that something could be. Ne-dominant types are the most creative and adaptive of all types. Se is also very adaptive, spur of the moment, but Ne is adaptive in a timeless sense. If things change suddenly or unexpectedly, the odds are that Ne has seen the change as something that could happen, and the person has wondered what they would do in that situation. They can be startled, of course, but it's quite an impressive feat to truly surprise someone with a strong Ne!
If you are an intuitive, you will probably understand how Ne can be called a "divergent" function. It looks at one situation, and imagines many possibilities. It looks at one object, and invents many uses. It starts by looking at banks, and ends up filtering through politics, conspiricies, political theory, philosophy, and then many individual philosophers' metaphysical ideas. Wikipedia is truly Intuition's Playground. In each of these cases, however, Ne begins at one point and splits off into many directions.
Intraverted Intuition - Ni
Ni is in intraverted Perceiving function, meaning that, like Si, it gathers information from within. Specifically, it perceives abstract meanings, symbols, underlying principles, and connects wildly varying ideas from a single root. For this reason, many Ni-dominant people feel like aliens, as if they perceive a completely different reality from everyone else. As they begin to grow up, they may become confused and wonder why everyone else doesn't see things as interrelated as they do. Ni is also the function of double meanings. While an Ne-person may argue two contradicting points of view at two separate times, Ni will--sometimes quite reasonably--affirm both points of view at the same time, and honestly believe that both are true. Ni is also very good at gathering the information necessary to predict the future, and seeing a situation from different perspectives. A person with a strong Ni can switch from one point of view to another almost effortlessly, flip flopping until a most favorable one--or most favorable few--are found. People with a strong Ni will find the deep themes of poetry as easy to interpret as the surface ones, and have the ability to easily pick skills learned in one area and reapply them in another.
For these reasons, Ni is a very "convergent" function. Contradicting ideas are brought together and called "same." The lines between different points of view are blurred. Many different situations are solved using the same set of internal principles. Many different symbols collide into the same physical object. While Ne diverges from one point into many, Ni synthesizes many points into one.