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Let's take it word by word:
Self: You or your organisation owns this thing. By contrast, a cloud-based offering by GitHub.com, Azure, GitLab.com, etc. (see this list on Wikipedia for a more comprehensive list) is not owned by you, but rather the service provider in question. Do not confuse ownership of the code (dictated by it's license – typically owned by you), with ownership of the service for performing version control on the code.
Hosted: You or your organisation are responsible for operating and maintaining this thing. In some cases, this is exactly the same software (e.g. one can self-host GitLab or run GitHub Enterprise “on premise” in your own datacentre). This means keeping it secure, making any changes to it, serving user requests for change, etc. You are also responsible for some part of the infrastructure necessary to provide this service. This may vary, depending on who you have service provision agreements with, from everything down to the power and networking, or only the last layer of the service (here, the version control system itself), or anything in between.
"version control system" (or VCS): anything (software or otherwise, but most likely software) which provides the functionality necessary to control versions. Note there is a subtle difference between Version Control and Version Control Software, although the meaning is often conflated.
The term VCS can refer to any version control system, even when it is not applied to software, however in our context, it can be assumed that you are referring to a version control system applied to software.
In other words, a “self-hosted VCS” can be thought of as:
A service and all of its components which is under your own ownership and control, acting as a version control system for software.
Often, this can take the form of:
base infrastructure (network/power/server),
operating system,
authentication and authorisation mechanism, and
the VCS server itself (Git, SVN, etc.).
The second part of your question is about how it is different from GitHub. The following contrast can be made:
Functionally there may be differences between what you can do with GitHub an what you can do with your self-hosted VCS. GitHub, as is the case for almost all VCS hosted by third parties (i.e., not hosted by you) goes a few steps further than simply doing version control. There are collaboration tools, identity management, issue tracking, etc. built into their service. While this is strictly speaking not part of VCS, it greatly improves the experience of using one, almost to the point where doing version control without them feels impossible.
GitHub owns and operates the service. You may make suggestions to change their service, API, etc., and perhaps even contributions to their open source products, but you do not own the service. It is provided to you based on the SLA for the offering that you choose. You cannot make your own changes to it. This is overall a very good thing, unless you are in the business of providing the VCS itself – but chances are that you just want to use the service, not compete with it.
Update to answer the edited question:
does it get uploaded to the server, or it is saved on my local machine? If it is uploaded to the server, then if I reset the computer, will I able to access my code again?
You are conflating two issues here: self-hosting and distributed version control. Of course, if you self-host on your own computer, your run the risk of losing everything if the machine dies. However, of someone else hosts your code in only one place, then you still risk losing everything if their machine goes down.
To address this (and several other issues related to centralisation of source control), distributed version control was invented. A good discussion is in the git book.
