How to integrate magento REST API with third party?

You can find a good explanation of the Magento REST API here. There is also an example on how to retrieve the products as a logged in customer. I will reproduce it here, to make the answer longer.

<?php
/**
 * Example of products list retrieve using Customer account via Magento REST API. OAuth authorization is used
 */
$callbackUrl = "http://yourhost/oauth_customer.php";
$temporaryCredentialsRequestUrl = "http://magentohost/oauth/initiate?oauth_callback=" . urlencode($callbackUrl);
$adminAuthorizationUrl = 'http://magentohost/oauth/authorize';
$accessTokenRequestUrl = 'http://magentohost/oauth/token';
$apiUrl = 'http://magentohost/api/rest';
$consumerKey = 'yourconsumerkey';
$consumerSecret = 'yourconsumersecret';

session_start();
if (!isset($_GET['oauth_token']) && isset($_SESSION['state']) && $_SESSION['state'] == 1) {
    $_SESSION['state'] = 0;
}
try {
    $authType = ($_SESSION['state'] == 2) ? OAUTH_AUTH_TYPE_AUTHORIZATION : OAUTH_AUTH_TYPE_URI;
    $oauthClient = new OAuth($consumerKey, $consumerSecret, OAUTH_SIG_METHOD_HMACSHA1, $authType);
    $oauthClient->enableDebug();

    if (!isset($_GET['oauth_token']) && !$_SESSION['state']) {
        $requestToken = $oauthClient->getRequestToken($temporaryCredentialsRequestUrl);
        $_SESSION['secret'] = $requestToken['oauth_token_secret'];
        $_SESSION['state'] = 1;
        header('Location: ' . $adminAuthorizationUrl . '?oauth_token=' . $requestToken['oauth_token']);
        exit;
    } else if ($_SESSION['state'] == 1) {
        $oauthClient->setToken($_GET['oauth_token'], $_SESSION['secret']);
        $accessToken = $oauthClient->getAccessToken($accessTokenRequestUrl);
        $_SESSION['state'] = 2;
        $_SESSION['token'] = $accessToken['oauth_token'];
        $_SESSION['secret'] = $accessToken['oauth_token_secret'];
        header('Location: ' . $callbackUrl);
        exit;
    } else {
        $oauthClient->setToken($_SESSION['token'], $_SESSION['secret']);
        $resourceUrl = "$apiUrl/products";
        $oauthClient->fetch($resourceUrl);
        $productsList = json_decode($oauthClient->getLastResponse());
        print_r($productsList);
    }
} catch (OAuthException $e) {
    print_r($e);
}

From the above code, you can token and token secret, just copy it:

...........
echo 'token:---'.$_SESSION['token'].'----secret----'.$_SESSION['secret'];
........

So, you can prepare a code like below to create/edit a product:

<?php
$apiUrl = 'APIURL';
$consumerKey = 'CONSUMERKEY';
$consumerSecret = 'CONSUMERSECRED';
$token = 'TOCKEN';
$tokensecret = 'TOKENSCRET';

try {

    $oauthClient = new OAuth($consumerKey, $consumerSecret, OAUTH_SIG_METHOD_HMACSHA1);
    $oauthClient->setToken($token, $tokensecret);
    $oauthClient->enableDebug();          

    $productData = json_encode(array(           
            'name'              => 'TEST PRODUCT',           
            'price'             => 11.11          
        ));       

    $resourceUrl = "$apiUrl/products/222";
    $oauthClient->fetch($resourceUrl, $productData , 'PUT',  array('Content-Type' => 'application/json'));
    $responseArr = json_decode($oauthClient->getLastResponse());
    print_r($responseArr);

} catch (OAuthException $e) {
    print_r($e);
}

?>