What is the process or api they use to provide this feature on their websites? If it is too complication to explain can you please push in the right direction to be able to provide this feature aswell. Thank you.
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Kasapo
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The key word you are looking for is geolocation for the first part, and proximity search for the second.
Google is the most prominent provider of Geolocation services, but there are others.
To get the latitude/longitude of an input address, see How to obtain longitude and latitude for a street address programmatically (and legally)
Here is one approach to do calculations on nearby locations: Get nearest places on Google Maps, using MySQL spatial data
And a very detailed question discussing the advantages and disadvantages of the various proximity formulas: Formulas to Calculate Geo Proximity