Here, intelligence is part of the brain designed to handle certain aspects. For example, there are people that cannot recognize faces, but can recognize other objects. Because there are enough of these people and we know what parts of their brains are damaged and we can see brain activity in those regions in well functioning people, we can assume that face recognition is an intelligence. It's a specific function the human brain is programmed to do.
These 7 intelligences are not the only types of intelligence, but they're the general types. Most humans have varying levels in each type. Some people are savants in specific categories and some lack even basic intelligence in a few categories. And even within this categories, it's possible to get more specific regarding brain functioning.
Body-Kinetic
This is in regards to how well we have the ability to control our body. It's one thing to know we want to move our right and one way or fingers another, but the intelligence (brain capabilities) is another. Surgeons would need great control with their fingers. Pianists, violinists and other musicians as well. Dancers and exceptional athletes. We all have basic capabilities, some are much better at it and some lack the basic functioning. This is beyond just physical capabilties, but how the brain processes whatever information your ears and eyes receives and how it can handle doing the task you set out to do.
Linguistic
When you're writing a poem, how easy it is to get the words you want? When making an extemporaneous speech, how free flowing do the words come while maintaining grammar, eloquence, etc. How easy is it to make word analogies or reason using words? How easy is learning grammar for your native language and foreign languages? There was a person that went to the UN and could speak 26 languages fluently. It's much easier to learn languages when immersed in multi-lingual societies. However, for some writers, poets, linguits and others, this capabilities comes much more naturally.
Visual-Spacial
When driving or going places, how well can you visualize where you going and have an internal map in your head? If you were an navigator on long voyages, this intelligence would be very useful. When painting and drawing, seeing colors and designs and rearranging shapes. When redecorating the room or reorganizing the fridge, knowing the shapes and sizes of objects and mentally manipulating them. This is a broader category then others, in my opinion.
Musical
I think we've all heard of musical savants. There abilities to manipulate notes and music in their heads at early ages. It's something I lack greatly. I know the words but barely understand what they mean: keys, melodies, etc..
Logical-Mathematical
This is well known and self-explanetory.
Interpersonal
How well can you notice people's moods and feelings? How well can you notice what someone is thinking through body language? How well can you deal and cope with others and their various attitudes? Car salesmen, politicians, confidants, etc. Some people are exceptional athletes, but have no ability to work in a team.
Intrapersonal
How well do you know yourself? How honest can you be with yourself? How much influence and control do you have over yourself? Some people are brilliant in math, but have little emotional control. They might go off on someone and not know why or how to control themselves.