'vein'. It's used when a character is angry or annoyed, so a small vein is displayed at the back of their head or at the front corner. If the character is more and more annoyed the more veins come up.
Another one is a 'nose bleed'. Mostly is used for male anime characters when they see see a pretty woman, or there's something perverted. Nose leaking from the nose. Sometimes is used for woman too. Often characters have a 'fountain' nose bleed as something happened unexpectionally and they fall on the ground.
If a person is complemented, says something cute or about their romantic feeling a small parallel lines appear underneath their eyes or a blush.
childish sometimes they will pull their eye lid down and show their tongue. It's mostly done by teenage girls towards boys, or elementary children when they are cheeky towards others (like classmates)
Orz isn't a facial expression. It's when a character kneels down or hide in the corner often while there's a navy/black backround with a spotlight to show that they are cruched by sudden event, they are defeated, they have no hope and they have given up. Also sometimes there are navy parallel lines bside it.
Another famous animation is 'sweat drop'. It's when a character/s have a drop on the back of their head or at the front corner to show that they are confused or suprised about an event happening. It's like saying 'What the hell?', 'Really?' ._. Also sometimes there are lots of small (sweat) drops coming through the top of their head (sometimes it's while they wave their hands and turn their head from one side to another) showing that they are panicing.
Another animation is when a character is suprised or shocked about a thing that just happened, so they perform a 'Face Fault'. It's when a character falls down on (sometimes straight on their face) and their legs are up.Or if their face is not as normal (deformed) I don't know which one is the right defenition :/
When a character is suprised or shocked they sometimes have a 'beady eyes'. It's when a character have a small circles instead of eyes which are white, while their mouth is wide open. It's like they are saying ' What?' Sometimes there's a question mark right beside their head.
fully white. This is used when a character/s is/are suprised or there is an awkward moment.
Nekko mouth is used for cutness and silliness of a character. Sometimes their hand is formed / done like cats (half-close their hands at put it beneath their chin)
Heart eyes - when a character has heart instead of eyes to show their love interest towards a character or object.
When a character sighs a cloud appears right beside their mouth which is from where coming from. Also their eyes are closed while doing this action that they are just fed-up.
Another animation is when from a character a 'small ghost' similar to them. Usually it is coming from their mouth. This is used when a character is fed-up or to give an impression that he/she died (in funny way).
'starry eyes' (stars in their eyes), showing that they adore something that is before them or if they are really determined to do something.
There are like dozens of other technques that are combined togather to fully show the expression of the character but if you want you can mention some in the comments! ^_^ The best person that show lots of expression is Tamaki Suoh form Ouran Highschool Host Club!