This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my textbook (Griffiths) and read countless of pdf's on the web but for some reason I am just not getting it.
Can someone explain to me, in the simplest terms possible, what this "Bra" and "Ket" (Dirac) notation is, why it is so important in quantum mechanics and how it relates to Hilbert spaces? I would be infinitely grateful for an explanation that would actually help me understand this.
Edit 1: I want to thank everyone for the amazing answers I have received so far. Unfortunately I am still on the road and unable to properly read some of the replies on my phone. When I get home I will read and respond to all of the replies and accept an answer.
Edit 2: I just got home and had a chance to read and re-read all of the answers. I want to thank everyone again for the amazing help over the past few days. All individual answers were great. However, the combination of all answers is what really helped me understand bra-ket notation. For that reason I cannot really single out and accept a "best answer". Since I have to accept an answer, I will use a random number generator and accept a random answer. For anyone returning to this question at a later time: Please read all the answers! All of them are amazing.
Best Answer
In short terms, kets are vectors on your Hilbert space, while bras are linear functionals of the kets to the complex plane
$$\left|\psi\right>\in \mathcal{H}$$
\begin{split} \left<\phi\right|:\mathcal{H} &\to \mathbb{C}\\ \left|\psi\right> &\mapsto \left<\phi\middle|\psi\right> \end{split}
Due to the Riesz-Frechet theorem, a correspondence can be established between $\mathcal{H}$ and the space of linear functionals where the bras live, thereby the maybe slightly ambiguous notation.
If you want a little more detailed explanation, check out page 39 onwards of Galindo & Pascual: http://www.springer.com/fr/book/9783642838569.