Nervous Wreck

There are some characters who can keep a cool head no matter what. Then at the other end of the scale, there's this person. They'll be the first to panic under pressure, and Heroic BSODs are a common thing for them. Pessimistic by nature, they'll automatically assume the worst and worry even when everything's going right.

Often, this character is revealed to have a painful past that left them like this. As such, this is sometimes a trait of a Shrinking Violet or a more emotional Broken Bird. Some versions of this will hide their anxiety (or at least try to) behind a cheerful facade.

Contrast The Stoic, Nerves of Steel.

Examples of Nervous Wreck include:

Anime and Manga

Miranda in D.Gray-man.
Joe from Digimon Adventure, although he mellowed out in Digimon Adventure 02, as he matured.
Chrona from Soul Eater, thanks to being raised to become a new Kisin by Medusa. S/he's even quite jittery when s/he's Axe Crazy.

Fan Works

In Dragon Ball Abridged, this happens to Krillin. He cannot shut up when he's scared.

Film

Woody Allen is famous for playing this kind of character.
Panic in |Hercules. The series also introduced Neurosis, whom even Panic considers a mess.
Ken the stutterer in A Fish Called Wanda.
In the Star Wars Universe, Nute Gunray (the Trade Federation Viceroy) nearly embodies this trope. Although beings from his species, Neimodians, are usually pessimistics and prone to great stress by nature, Gunray overly nervous personnality is well beyond even Neimoidian standards.
Leo Bloom in all adaptations of The Producers.
Felix in the film version of The Odd Couple.

Literature

Nettie in Needful Things is always tense and jumpy due to the abuse she suffered at the hands of her husband. Keeton also gradually becomes this as paranoia sets in.
The Bursar from Discworld books. He starts off fine in Eric and becomes only slightly twitchy in Moving Pictures. However the events of that book, and the following one Reaper Man leave him a paranoid, twitching, nervous mess, who has to be medicated into hallucinating he is sane (attempts to cure of his nervous state proved impossible).
In The Dresden Files, Molly Carpenter becomes this in Ghost Story, following Harry's apparent death in the previous book and her own attempt to fill in his shoes as the magic defender of Chicago.
Piglet, from Winnie the Pooh, is a classic example: meek, timid, stuttering, always fretting over something, and in the Disney version is usually seen visibly trembling.
The White Rabbit, in almost all adaptations of Alice in Wonderland, is a Nervous Wreck due to being "late for a very important date," even going so far as twitching a lot.
Shrewtooth from Warrior Cats.

Live-Action TV

Doug from Scrubs, nervous about...everything.
These show up a LOT in Canada's Worst Driver. They're as scary as road ragers and ditzes.
Rebecca from Cheers.
Miles from Murphy Brown.

Music

"Basket Case" by Green Day is from the point of view of an extremely neurotic person.

Newspaper Comics

One character in Dilbert named Ted (not to be confused with Ted the generic guy) with an appropriate nickname.

Video Games

In The Sims 3, any Sim with the Neurotic trait is this.
Hanako of Katawa Shoujo has severe anxiety issues and regularly has panic attacks, and her teacher knows to simply let her run out of class if she needs to.

Web Comics

Dies Horribly in Goblins Life Through Their Eyes is always on edge. This is justified because most Goblins in his clan are named due to prophecy.
Jack Delitt from Newheimburg has this issue in spades.

Web Original

Edward Borman in The Mercury Men. Justified, given the situation he's in.
The Nostalgia Critic is very prone to panic and stressing out over nothing.

Western Animation

Tweek from South Park.
The Earl of Lemongrab from Adventure Time. Whether his character and situation are hilarious, or terribly depressing, is debatable.
In the episode "The Truth Hurts" from The Replacements, the head of the school newspaper who is replaced is a Nervous Wreck who is constantly jittery and gets extremely nervous about the idea of anything being published in the school newspaper that would make things like fruit on the bottom yogurt as opposed to being like plain yogurt, as he likes it.
Shaggy from Scooby Doo is usually this.
Rainbow Dash from My Little Pony Friendship Is Magic normally presents the exact opposite of this. However, in a few cases it's shown that her cocky facade is just that, and when she is no longer able to put up that facade she collapses into a ball of nerves.
Jitters A. Dog from Raw Toonage and Bonkers.
Mr. Nervous from The Mr. Men Show. He frequently lapses into daydreams, sort of blown-out-of-proportion scenarios involving whatever is going on around him at the time. Whether it be a short two-minute ride outside a store or a spider crawling out of his musical instrument. He will then usually flee for his life, screaming and shouting 'Oh no no no!' or 'This is the end!'

Real Life

Generalized anxiety disorder.
Some people with PTSD are this to some measure. Because of their trauma, they are constantly "on guard" and wary of something bad happening to them. Additionally, they may be easily startled and have a Hair-Trigger Temper.