Even though today it is very difficult to separate the internet from the World Wide Web, there were many other services that were running over the internet or other networks prior or simultaneously to the web. Most of these remain as small reducts for true hobbyists, while others have disappeared completely and others are still in full use. These are four things that you were able to do before the web even existed.
Bulletin Board System (BBS)
The idea behind the BBS was to have an online way to communicate through messages, in the same way that we do these days with online forums. The first BBS software was created in 1978 by Ward Christensen. These systems were in use mostly during the 80s and early 90s, when other types of communication started to take the users away from them. Essentially, they were text-only interfaces with menus. Navigating those menus you could post messages to other users, read the messages that others posted publicly or privately and, in the most advances systems, upload and download files. The BBS systems were accessible through dial-up connections on the early times, and after that, through the internet as well. These systems became, during the peak of its usage, the center of file sharing, being part of the golden era of shareware. Shareware is a type of software that is free to share and try, and after that you are supposed to pay the author to get a license or the full product. If you are interested on watching how this used to work, the Telnet BBS Guide is a good resource to start.
The Users Network (Usenet)
Usenet was created in 1979 by Tom Truscott and Jim Ellis from Duke University in the USA, though it started to work in the wild in 1980. The system allows the users to exchange opinions and pieces of information with other people interested on the same topics than him, by submitting "articles" to different interest groups organized hierarchically and called "newsgroups". For example, the comp.ai group discusses artificial intelligence, while the comp.theory talks about theoretical computer science. Both groups are under the hierarchy comp, which means that they are related to computers or computer science. This system could be understood as a mix between email and forums, with the ability to upload and download binary files. Nowadays there are many providers that give access to the Usenet through the Internet. Usually, it is a paid service.
Electronic Mail (Email)
Email, as we can recognize it today, started to be used in the mid-1970s, as a mean for the users to communicate while using time-sharing computers, in which each user had slots of times to execute programs. In the beginning, both users needed to be online at the same time for the email to reach its destination, making it some sort of instant messaging system, but later on the service changed to what it currently is. Initially, messages were text-only, but promptly the ability to send files was added. These days we are extremely used to read and write email addresses, but the first documented use of the "at symbol" (@) for the purpose of indicating the server to which a user belongs took place in 1971 when Ray Tomlinson was creating the specification. Previously, you could only send emails to another user inside the same computer you belonged to. For the design and initial implementation of Electronic Mail, Tomlinson was awarded several prizes, including the IEEE Internet Award and an induction in the Internet Hall of Fame.
Gopher was presented in mid-1991 by a team lead by Mark P. Cahill at the University of Minnesota. The system is organized as hierarchical menus that contain documents that can be read by using a client. Originally intended as an alternative to the World Wide Web that enforces more strictly the structure and organization of the documents, it soon became mostly unused when it became ubiquitous. That said, there are still many hobbyists trying to revitalize the system, which has made the number of Gopher servers grow during the last decade according to Veronica-2, the Gopher search engine. Most modern browsers have dropped the support for this protocol.