How to Integrate 3rd Party Shipping API in Magento 2.1

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);
}

In Magento 2 you have to create custom JavaScript component responsible for redirect to a Payment Provider. So how it works in Magento 2.

Custom redirect.html template with "Place Order" button should be created This template is controlled by a redirect.js JavaScript component with afterPlaceOrder function inside which you can send HTTP GET request to your custom controller which will return parameters required for sending HTTP POST request to a payment gateway.

afterPlaceOrder: function () {
    var self = this;
    $.get(config.getDataUrl())
        .done(function (response) {
            customerData.invalidate(['cart']);
            formBuilder(response).submit();
        }).fail(function (response) {
            errorProcessor.process(response, self.messageContainer);
        }).always(function () {
            fullScreenLoader.stopLoader();
        });
}

The config.getDataUrl() is custom controller URL. Here you do implement 2 different ways of POST parameters preparation. URL can be prepared from backend side and provided to a frontend by implementing the \Magento\Checkout\Model\ConfigProviderInterface::getConfig() method.

Return key => value array of parameters as a response. The formBuilder(response).submit() will foreach parameters and prepare form which will be submitted to a payment provider as post.

var formTmpl =
    '<form action="<%= data.action %>" method="POST" hidden enctype="application/x-www-form-urlencoded">' +
    '<% _.each(data.fields, function(val, key){ %>' +
    '<input value="<%= val %>" name="<%= key %>" type="hidden">' +
    '<% }); %>' +
    '</form>';

return function (response) {
    var inputs = {};
    for (var index in response.fields) {
        inputs[response.fields[index]] = response.values[index]
    }
    var hiddenFormTmpl = mageTemplate(formTmpl);
    var tmpl = hiddenFormTmpl({
        data: {
            action: response.action,
            fields: inputs
        }
    });
    return $(tmpl).appendTo($('[data-container="body"]'));
};

Render form wrapped in a <html><body><form/></body></html> on backend side using Block and *.phtml file with inline JavaScript inside the response. Once retrieved, JavaScript will trigger redirect. It is less recommended way but might be less time consuming.