Here's just some of the many logos I've designed for various clients over the years. I try to create logos that feature some impactful use of negative space or other eye catching elements. Crosskeys Books for example, are a pet related bookshop, so in their logo you might see the pages of a book or the whiskers of a cat. Cryptocard, the authentication experts, help to ensure websites are locked down and accessible only to those with authenticated access. So in their logo, you might see the shape of a keyhole as in the negative space of the second C.
I usually sketch out ideas for logos before bringing those into Illustrator.