This is a very broad question, and certainly too broad to be comprehensively answered here. Broadly though you can consider the following steps as important in speech recognition/natural language processing (NLP).
1) Analog to digital conversion. This encompasses issues such as sampling the speech, filtering/noise reduction, and segmentation of the signal into chunks that can be processed.
2) Representing natural language. That is, somehow modelling the language so that you can understand how it is constructed. Typically for this both language and acoustic models are used (https://www.youtube.com/watch?v=K1tBjg503uU). These are statistical representations of common patterns and models of structures in language.
3) Infering the most likely translation. This is the process of actually 'translating' the speech to text (or whatever output you expect from your speech recognition). Typically a machine learning/statistical step. Unsurprisingly numerous algorithms have been employed to achieve this, with the most prominent that I have seen being the hidden Markov Model (https://www.youtube.com/watch?v=TPRoLreU9lA).
As I say, this is very broad and there are a number of issues to deal with. For a good introduction to the latter 2 issues I highly recommend taking a free course such as the NLP course on Corsera
You should also note that NLP is a computationally difficult problem and one that is an active area of research spanning a number of fields (e.g computing science, linguistics, mathematics, statistics). Therefore, if you intend to seriously understand contemporary methods then you should be prepared to do a lot of reading - as there is no easy solution. That being said, there are plenty of kind people out there who have helpfully made excellent software that can be used (e.g http://cmusphinx.sourceforge.net/). So with a basic grasp of the issues I mentioned you should be able to use such software effectively.
Best of luck in delving into what is a challenging but fascinating area of active research.