Twig

Twig is a modern PHP-based compiled templating language. It is part of the Symfony 2 framework and is the default template engine in Drupal 8 (Read the official change record).

All of the theme_* functions and PHPTemplate based *.tpl.php files have been completely removed and replaced with Twig template files. These template files have a new Twig extension: *.html.twig.

When your web page renders, the Twig template engine takes the template file and converts it into a 'compiled' PHP template which it stores in a protected directory in sites/default/files/php_storage/* The compilation is done once. Template files are cached for reuse and are recompiled on clearing the Twig cache.

Coding standards

Twig is a new core programming language and all of the core template files must follow the coding standards. There are both official coding standards from the Twig template engine and coding standards for the usage of Twig in Drupal.

Generic Twig coding standards: http://twig.sensiolabs.org/doc/coding_standards.html
Twig in Drupal coding standards: http://drupal.org/node/1823416

Advantages

Compared to PHPTemplate, Twig offers some major advantages. Especially when it comes to security.

Twig is more secure, due to the fact that only a number of tags can be used. In the previous PHPTemplate, it was possible for a template file to execute the following code:
```
  <?php
    db_query('DROP TABLE {users}');
  ?>
```
This should of course not be the case. In case you're wondering what it does: it removes the entire users table from your database. Not good, right?
There is now a clear separation between the logic and the view. This means: no more PHP code inside your template files.
The syntax is very easy to understand, making the code more readable as well. Also, many IDE's have syntax highlighting for *.twig files.
Template files are reusable, thanks to Twig includes.
Debugging is much more easy. First of all, Twig outputs a helpful message with the filename and the line number whenever there is a syntax problem within a template. Secondly, you can turn on a Twig debug function. More on that later.
Twig is very well documented. Go ahead and start reading the official documentation here.
It's not only used in Drupal core, so it's not a Drupaly-thing.

Disadvantage

Although the syntax is very easy to read and understand, it's a new syntax you have to learn and get used to before getting started. Once you know the syntax however, you get the advantage of knowing a template engine that can be used outside of Drupal as well. This is due to the fact that Twig is part of the Symfony framework.

Read more

The official Twig documentation from Sensio Labs.