In order to get the base URL of the REST API for your account, log in your account and look for it inside the Settings/Web Services/API Explorer
It will look like
https://apiserver/ws/rest/api/version
When you are composing the URL of a resource, always start with the https://apiserver/ws/rest/api prefix, append the resource type (for example 'contacts' of 'simplemessages') and then the detailed operation path.
For instance, if you want to get data about the contact with id 987, your URL will look like
https://apiserver/ws/rest/api/contacts/contact/987
Authentication
In order to access the API you have to authenticate yourself, usually you will use the OAuth 2.0 protocol.
See OAuth documentation in order to learn how to get an access_token.
When you have got an access_token you can use it to call this API.
Using the access_token in the querystring
The most simple way is to append an "access_token" parameter to the query string
GET https://apiserver/ws/rest/api/contacts/contact/987?access_token=XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
Using the Bearer HTTP Header
Another simple way is to add an HTTP header Bearer to your request
GET https://apiserver/ws/rest/api/contacts/contact/987 Authorization: Bearer XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
This is the preferred way, because HTTP headers usually are not logged and so your access_token is never recorded anywhere.