Search Network: These are the ads that show up in the Google search engine. They are only available in text format.
You would choose this if you wanted your ads to come up as people are searching for your specific product or service.
Display Network: Your ads are matched to websites that have content relevant to your ads (which are matched through the keywords).
You can also choose which websites they are featured on, what topics it is placed around, and the demographic of viewers. They are displayed in text, image, rich media and video.
These are displayed on the Google Network, which are websites who host Google ads.
You would use this to build awareness around your brand, product or service across a large audience.