From Project Gemini:
Gemini is a new, collaboratively designed internet protocol, which explores the space inbetween gopher and the web, striving to address (perceived) limitations of one while avoiding the (undeniable) pitfalls of the other.
To put it into my own words, Gemini is a text-only web protocol built as a reaction against the modern web. But what is wrong with the modern web? Let me answer that with a few of Drew DeVault's blog posts:
Okay, so you have some context. What now? Since Gemini is a separate protocol from www (the "web"), you need to access it in a different way. There are two main ways I do this right now: