Array of objects

You could use filter:

function findLang(arr, code) {
  var filtered = arr.filter(function (el) {
    return el.lang_code === code;
  });

  // filter returns an array of objects that match the criteria
  // if the array is not empty return the language,
  // otherwise return 'invalid'
  return filtered.length > 0 ? filtered[0].name : 'invalid';
}

findLang(arr, 'eng'); // English

DEMO

If you wanted to add map into the mix instead of using that ternary operation (but which would most likely be slower and doesn't really provide any additional benefit):

function findLang(arr, code) {
  return arr.filter(function (el) {
    return el.lang_code === code;
  }).map(function (el) {
    return el.name;
  })[0] || 'invalid';
}

DEMO

How about a for loop. Something like:

function find_lang(input_arr, lang_code) {
   for(var i = 0; i < input_arr.length; i++) {
      var o = input_arr[i];
      if( o.lang_code === lang_code ) {
         return o.name;
      }
   }

   return "invalid";
}

Underscore.js has some useful helpers for this kind of thing: http://underscorejs.org/

E.g Find:

Looks through each value in the list, returning the first one that passes a truth test (predicate), or undefined if no value passes the test. The function returns as soon as it finds an acceptable element, and doesn't traverse the entire list.

var even = _.find([1, 2, 3, 4, 5, 6], function(num){ return num % 2 == 0; });
    => 2

Where:

Looks through each value in the list, returning an array of all the values that contain all of the key-value pairs listed in properties.

_.where(listOfPlays, {author: "Shakespeare", year: 1611});
=> [{title: "Cymbeline", author: "Shakespeare", year: 1611},
    {title: "The Tempest", author: "Shakespeare", year: 1611}]

You could do it this way:

// configure your parameters
function checkArr(){
    for (var i = 0; i < x.length; i++){
        if (x[i].lang_code == "eng")
            return "English";
    }
    return "invalid";
}

var x = [
{
    "lang_code":        "eng",
    "site_language":    "1",
    "name":         "English"
},
{
    "lang_code":        "afr",
    "site_language":    "1",
    "name":         "Afrikaans"
},
{
    "lang_code":        "ale",
    "site_language":    "0",
    "name":         "Aleut"
}
];