This is quite a question...
In a nutshell, such systems measure the accelerations they are exposed to and use those values to deduce changes to their position in space. But trying to briefly explain it on here would not do these systems, their complexity, their inherent errors and the different installations in various aircraft types justice at all.
IŽd suggest you hit the books and obtain a basic understanding of gyros and the effects of accelerations on the same first. Then go on with your course documentation on dead reckoning navigation and finally read the same documentation on the inertial navigation systems. They have several inputs they use for their internal number crunching. Look at the bunch of other accelerations beyond aircraft movement relative to the earth such a system is subject to and how they are eliminated out of the calculations. Then you will also see why and how the IRS is suffering from measuring errors, especially in the long run. Also consider the interaction of the inertial systems with other navigational sources and how they are used to cancel each others inherent weaknesses out.
And, that done, look at the individual type you are operating to see how exactly the systems work there and what the SOPs and abnormal procedures say on their use.