How to get visitor's location (i.e. country) using geolocation?

You can use my service, http://ipinfo.io, for this. It will give you the client IP, hostname, geolocation information (city, region, country, area code, zip code etc) and network owner. Here's a simple example that logs the city and country:

$.get("https://ipinfo.io", function(response) {
    console.log(response.city, response.country);
}, "jsonp");

Here's a more detailed JSFiddle example that also prints out the full response information, so you can see all of the available details: http://jsfiddle.net/zK5FN/2/

The location will generally be less accurate than the native geolocation details, but it doesn't require any user permission.

You don't need to locate the user if you only need their country. You can look their IP address up in any IP-to-location service (like maxmind, ipregistry or ip2location). This will be accurate most of the time.

Here is a client-side example with Ipregistry (disclaimer, I am working for):

fetch('https://api.ipregistry.co/?key=tryout')
    .then(function (response) {
        return response.json();
    })
    .then(function (payload) {
        console.log(payload.location.country.name + ', ' + payload.location.city);
    });

If you really need to get their location, you can get their lat/lng with that method, then query Google's or Yahoo's reverse geocoding service.