"Jailbreak" is a term that generally refers to removing the built in restrictions on an iOS mobile device, granting access to the root file system and root user privileges. The equivalent on Android devices is often called "rooting".
Jailbreaks use existing holes in iOS's security to grant the user root access. Since iOS isn't designed for this, but rather designed to avoid it, jailbreaking a device will usually void the warranty. 
A jailbreak represents a vulnerability in iOS, and a jailbreaked system is significantly more vulnerable. This is also a major factor in why your ddevice's arranty will generally be voided if you jailbreak it.
Different jailbreakers use different exploits and vulnerabilities to jailbreak a device, so it is difficult to say how a specific one is does this. Also, since Apple actively tries to remove vulnerabilities, jailbreaks will be different depending on the iOS version. This also means that you generally can't update a jailbreaked device.
If you want to know what a specific jailbreaker does, I suggest looking at their documentation. For checkra1n, that would be: https://github.com/checkra1n
If you have any other questions regarding jailbreaks,or another topic, feel free to ask them.
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