If you want your server to authenticate with some fixed credentials, then you have 2 choices.
You can use the OAuth2 username/password flow, this uses the OAuth infrastructure, but allows you to programmatically get an access token given a username, password (and possibly security token), this is a simple POST request to the correct URL with all the parameters. see the online help for details.
You can use the login method in the Partner or Enterprise APIs. these will return you a sessionId, that you can then use in the REST API requests in exactly the same way you'd use an access token.
In either case, if your PHP server's IP address is not whitelisted, then you'll need to supply the users api security token in addition to the password.
Once you've gotten a token from either of these methods, you would use it the same way as a regular OAuth token, by putting it in the Authorization HTTP header of your Apex API request, e.g.
Authorization: Bearer {someLongTokenStringHere}
I would recommend you go with (1) unless you're already using the PHP toolkit to call the soap APIs.