Do you know the difference between the internet and the web? (This may seem like a pedantic distinction, as many people use the terms interchangeably; it is a necessary distinction, however, because gemini is on the internet but very much not on the web.)
The internet is a global networks of computers ("computers" including phones), which allows all of those computers to communicate with each other. It is the infrastructure that allows one computer to send data to another, but it does not specify in what form the data should be sent and interpreted.
The web is one particular thing that happens on the internet. Specifically, it is the communication that happens via HTTP, the "hypertext transfer protocol". The basic functionality of HTTP is a way for one computer, the "client", to request a file (a webpage, for example) from another computer, the "server", and for the server to send that file to the client.
HTTP is just one internet protocol, one form in which data is sent and received on the internet. E-mail, for example, is mostly transported via a different protocol.
Gemini is an internet protocol which allows a "client" to obtain a file from a "server" (just like HTTP). It comes with a document format, gemtext, which allows documents to contain "links" to other files (just like HTML).
Gemini is much younger than HTTP (2019 vs. 1991) and does far, far less.
Even the first version of HTTP could already do much more than gemini can, which begs the question: what is the point of making a new internet protocol which brings zero new functionality? The point is in fact to have less functionality.
The thing is: there are some downsides to the number of features that HTTP and HTML+CSS+JavaScript have. The main one is that a fully-featured webbrowser is an extremely complex piece of software. As a result, there are basically 3 commonly used webbrowser implementations and creating a new one from scratch is considered near-impossible. This gives the people in control of those 3 browsers quite a lot of power over their users, and it just so happens that the most popular webbrowser is owned by an evil, data-hungry advertising company.
Meanwhile, the gemini protocol is about as close as you can get to "client sends URL, server sends file, done", and the gemtext format is also pretty simple to interpret and display. Of course a fully-featured gemini browser does a bit more than just that, but with how many gemini browsers already exist, no one can use their browser's market share to push for protocol changes, and no browser author would dare introduce user-hostile antifeatures.
HTTP also allows website owners to collect lots of information on their users, through the use of cookies and a host of other techniques. On gemini, the potential for tracking is much more limited.
Gemtext is much simpler than HTML, and is geared more towards written documents. This is definitely a limitation, in the sense that there is a lot less you can do in a gemtext document, and gemtext pages can be relatively unappealing visually. The advantage is that there is a lot less clutter possible. Gemtext also avoids the tendency of the web to try and fit everything into a single format. You can find webpages that pull in an entire PDF reader implemented in JavaScript, just so a PDF document can be displayed on the webpage itself, instead of letting the user download the file and open it in a PDF reader of their choice. On gemini, site authors are only given the latter option.
Compared to a typical webpage, a gemtext document containing the same information will be much smaller in terms of data that needs to be transported. This is good, because transporting data over the internet takes energy.
You can install a gemini browser. There are quite a few to choose from (Wikipedia has a list); a popular one to get started with is Lagrange.
Alternatively, you can use a web-into-gemini proxy like portal.mozz.us .
Having taken Latin in secondary school, I personally pronounce it the same as the English words "gay me knee". But I seem to be in the minority on this.