It comes in tons of different forms - The one primarily being discussed here is called "Grapheme" in which letters/numbers/words 'have colors'. To my knowledge there are over 60 'types' of synesthesia - some of them occurring in pairs or individually. Grapheme is one of the more common ones. So is seeing colors when you hear sounds/music (For example, John Mayer claims that he experiences colors when he hears music). Sounds can have colors. You can experience colors when you feel particular emotions. Personalities or people can have colors. Flavors can have colors, colors can have flavors. Pain can have colors, pain can have flavors, etc. Basically take any sensation, couple it with a sense such as taste, touch, visual, etc.
Everyone probably has it to some degree. Even if you feel like inanimate objects have personalities - If you 'sense' that your car or you computer has a personality - that's synesthesia.
There's even a bit of 'cultural' synesthesia ingrained in us all. Things like assigning gender to objects or colors. Why is pink feminine and blue masculine to so many people? There's no real logical explanation for us to match a personality to a color, except that our brain has been taught to make that connection. That's sort of a taste of how synesthetes connect things, only they don't have to have it 'taught' to them. Yellow might be playful, green might be rude. It just is. They don't necessarily know why. But it is.