Why ket and bra notation? So, I've been trying to teach myself about quantum computing, and I found a great YouTube series called Quantum Computing for the Determined. However. Why do we use ket/bra notation? Normal vector notation is much clearer (okay, clearer because I've spent a couple of weeks versus two days with it, but still). Is there any significance to this notation? I'm assuming it's used for a reason, but what is that reason? I guess I just don't understand why you'd use ket notation when you have perfectly good notation already.
 A: I think there is a practical reason for ket notation in quantum computing, which is just that it minimises the use of subscripts, which can make things more readable sometimes.
If I have a single qubit, I can write its canonical basis vectors as $\mid 0 \rangle$ and $\mid 1 \rangle$ or as $\mathbf{e}_0$ and $\mathbf{e}_1$, it doesn't really make much difference. However, now suppose I have a system with four qubits. Now in "normal" vector notation the basis vectors would have to be something like $\mathbf{e}_{0000}$, $\mathbf{e}_{1011}$, etc. Having those long strings of digits typeset as tiny subscripts makes them kind of hard to read and doesn't look so great. With ket notation they're $\mid 0000\rangle$ and $\mid 1011\rangle$ etc., which improves this situation a bit. You could compare also $\mid\uparrow\rangle$, $\mid\to\rangle$, $\mid\uparrow\uparrow\downarrow\downarrow\rangle$,  etc. with $\mathbf{e}_{\uparrow}$, $\mathbf{e}_{\to}$,  $\mathbf{e}_{\uparrow\uparrow\downarrow\downarrow}\,\,$ for a similar issue.
A: All the answers so far provide valid reasons for Dirac notation (bra's and ket's). However, the central reason why Dirac felt the need to introduce this notation seems to be missing from these answers.
When I specify a quantity as a vector, say
$$ \mathbf{v}=[a, b, c, d, ...]^T $$
then in effect I have already decided what the basis is in terms of which the quantity is expressed. In other words, each entry represents the value (or coefficient) for that basis element.
When Dirac developed his notation, he realized that a quantum mechanical state contains the same information regardless of the basis in terms of which the state is expressed. So the notation is designed to represent this abstractness. The object $|\psi\rangle$ does not make any statement about the basis in terms of which it is expressed. If I want to consider it in terms of a particular basis (say the position basis) I would compute the contraction
$$ \langle x|\psi\rangle = \psi(x) $$
and then I end up with the wavefunction in the position basis. I can equally well express it in the Fourier basis 
$$ \langle k|\psi\rangle = \psi(k) $$
and obtain the wavefunction in the Fourier domain. Both $\psi(x)$ and $\psi(k)$ contain the same information, since they are related by a Fourier transform. However, each represents a certain bias in the sense that they are cast in terms of a particular basis. The power of the Dirac notation is that it allows one to do calculations without having to introduce this particular bias. I think this is a capability that Dirac notation provides that is not available in ordinary vector notation.
A: First, this notation makes it very clear which objects are interpreted as elements of the primal space (kets) or elements of the dual space (bras).
The names "bra" and "ket" recall how the notation was formed: as the left and right halves of an inner product, the projection of the state $a$ along the measurement $\phi$ is the inner product(indicated by angle brackets) $\langle \phi , a \rangle$, which can be typographically broken into $\langle \phi \mid\,\mid a \rangle$, two objects which typographically hint vector-ness.
There is also a typewriter limitation that contributes to this notation (and rather too many notations that are just variants of two or more elements in a comma-separated list bounded by parentheses or square brackets: GCD, LCM, object generated by, meet, join, intervals with various endpoint conventions, sequences, object tuples, et al...).  It is very time consuming to type a column vector on a typewriter.  Typewriters do not have oversized parentheses for column vectors.  This leads to strictly malformed constructions like "Let $A$ be a linear operator between $\Bbb{R}^2$ and $\Bbb{R}^2$, then $A \cdot (1,0)$ has ..." where a row vector is typed in a place that a column vector is required.  In particular, this means that the most common form of vector in beginning linear algebra is hard to typeset and so was frequently typed incorrectly transposed.
Further, elements of the primal and dual spaces should be readily distinguished (to prevent unintentionally writing, e.g., $\sum_i \mid \mathrm{e}_i \rangle \mid \mathrm{e}_i \rangle$).  However, the "obvious" solution is even harder to type: "$\sum_i \langle \mathrm{e}_i \mid \underline{\widehat{\mathrm{e}_i}}$" (and  even with the full power of MathJax, as much time as I'm willing to spend on this necessarily has the primal vector pointing up instead of down).
Finally, the stuff one puts in a bra or ket is seldom a set of vector components.  By the definitions that a mathematician uses, the components of a vector all come from the same field.  This isn't going to work for states described by some continuous and some discrete variables, or by states with some variables in the primal space and some variables in the tangent space.  (If we force this to work, we actually get direct sums of modules, not vector spaces.)  So while we might like to put lists of state-describing numbers in a bra or a ket, the thing we get is not and cannot be a (formal) vector.
A: The bra-ket notation is an advancement of the dot product of "normal" vectors.
$$ \vec{a} \cdot \vec{b} = \sum_i a_ib_i . $$
 This is generalized to the inner product $ \langle a, b \rangle $, which for functions is defined as:
$$ \langle a(x), b(x) \rangle = \int a(x)b(x) dx $$ 
in the simple case of 1-dim functions. 
Well the big advantage of the bra-ket notation is that there is no need to specify the representation, i.e. the coordinate system until one wants to calculate something in a specific space.

Part of the appeal of the notation is the abstract representation-independence it encodes, together with its versatility in producing a specific representation (e.g. x, or p, or eigenfunction base) without much ado, or excessive reliance on the nature of the linear spaces involved.

It is pretty handy when, for example, evaluating equations like
$$ \langle \psi_0 | ( |\psi_0\rangle + |\psi_1\rangle) = \langle \psi_0 |\psi_0\rangle ,$$
where $ |\psi_i\rangle $ are some orthogonal states. It is fast evaluation without the need to specify the system of $|\psi\rangle$ - whether $|\psi_0\rangle = (1, 0) $ and $|\psi_1\rangle = (0, 1) $  or it is $ |\psi_0\rangle = (1, \pi/2) $ and $|\psi_1\rangle = (1, 0) $ in polar coordinates $r, \varphi$. 
I do see, your point in saying "normal" vector notation is much clearer. That might be the case for these simple vectors as the ones above, but makes things hard to write when it comes to functions in multi- or even infinite-dimensional Hilbert space.
A: Indeed, I agree with you, standard notation is, in my personal view, already sufficiently clear and bra-ket  notation should be used when it is really useful. A typical case in QM is when a state vector is determined by a set of quantum numbers like this $$\left|l m s \right\rangle$$
Another case concerns the use of the so-called occupation numbers $$\left|n_{k_1} n_{k_2}\right\rangle$$ in QFT. Also q-bits notation for states $\left|0\right\rangle$, $\left|1\right\rangle$  in quantum information theory is meaningful...
Finally the use of bra ket notation permits one to denote orthogonal projectors onto subspaces in a very effective manner
$$\sum_{|m|\leq l}\left|l m  \right\rangle \left\langle l m\right|\:.$$  
A reason for its, in my view, nowadays not completely justified use is historical and due to the famous P.A.M. Dirac's textbook. In the 1930s, mathematical objects like Hilbert spaces and dual spaces, self-adjoint operators, were not very familiar mathematical tools to physicists.  (The modern notion of Hilbert space was invented in 1932 by J. von Neumann in his less famous textbook on mathematical foundations of QM.)
Dirac proposed a very nice notation which embodied a fundamental part of the formalism. However it also includes some drawbacks. In particular,  manipulating non-self adjoint operators, e.g., symmetries, turns out to be very cumbersome within bra-ket formalism. 
If $A$ is self-adjoint, in $\left\langle \psi\right| A\left| \phi\right\rangle$ the operator can be viewed, indifferently, as acting on the left or on the right preserving the final result. If the operator is not self-adjoint this is false. 
I think bra-ket notation is a very useful tool, but should be used "cum grano salis" in QM. In my view $\left|\psi\right\rangle$ where $\psi$ is a qunatum mechanics wavefunction, may be a dangerous notation, especially for students, as it generates misleading questions like this, $A\left|\psi\right\rangle = \left|A\psi \right\rangle$?
ADDENDUM. I understand that I interpreted the question into a broader view, regarding the use of bra-ket notation in QM rather than the restricted field of  quantum information theory.
A: The bra-ket notation comes from Dirac.
Feynman gives a good explanation in his Lectures on Physics, vol. 3, p 3-2.
If you are familiar with conditional probability, we write the probability of seeing $b$ if we have seen $a$ is written
$$P(b|a)$$
In quantum mechanics the calculation of seeing $b$, if we have already seen $a$, is written in bracket notation:
$$\langle b|a \rangle$$
which is the same idea, except that it is not a probability, but a complex number called a probability amplitude. In quantum mechanics we don't work with real numbers; the probability calculations give good predictions only when we work with complex numbers. At the end of calculations, the length of a complex number is squared to obtain the real-number probability we expect to observe:
$$| \langle b|a \rangle |^2$$
Now we can talk about posterior conditions and prior conditions as a "bra" $\langle b|$ and a "ket" $|a \rangle$.
Then if in place of a specific outcome $b$ we consider all possible outcomes, that is a vector, a "bra-vector".
The space of prior values (or states) is a "ket-vector".
A: What is "normal vector notation"?  I've seen angle brackets with commas, parentheses, square brackets, $\hat{x}$, $\hat{i}$, column matrices, row matrices ... which of those is "normal",  $(x|y)$, ...?    
Bras and kets are just another, with the particular benefit that it distinguishes the vector space from its dual space.  
edit after comment
Note that some of these are component notations, which do not work for quantum mechanics as the number of dimensions can be large or infinite.
A: The preference for bracket notation might be related to how make an elegant classic interpretation of quantum measurement.
Consider a system described by the state $|\beta\rangle$ then the average or expected value of an operator $\hat{A}$ which corresponds to the classical theory is simply the bracket of the operator. Or sandwiching the operator:
$$\langle \beta| \hat{A} |\beta\rangle$$
In the case of the hydrogen atom, for example, the bracket of position operator, for an electron in an eigenstate $|\epsilon_{n}\rangle$, is zero. Classically, therefore, electron is in the nucleus or origin:
$$\langle \epsilon_{n}| \hat{X} |\epsilon_{n}\rangle=0$$
Makes sense, classically, why no radiation is emitted while the system is in an energy eigenstate.
A: Here's an example. Let's say you're working with the free particle in introductory quantum mechanics, where the "vector" $ \vec{\psi} $ has infinitely many components (I know that sounds crazy if you don't have a lot of experience with quantum mechanics, but it's the case). With traditional notation, you can't keep track whether $ \vec{\psi} $ belongs to the dual or regular space - whether $ \vec{\psi} $ is a row vector or a column vector, respectively. In standard notation you'd have to write out the components (infinitely many of them!) to demonstrate a row or a column.
Bra-ket notation is nicer there. The "bras" $ \left \langle \psi \right | $ are dual vectors to the "kets" $ \left | \psi \right \rangle $.
A more crazy and more useful interpretation is that bras are linear functions and kets are their arguments.
