Brane
Definition: In theoretical physics, a brane (short for membrane) is an object which can have any number of allowed dimensions. Branes are most popular for their presence in string theory, where it is a fundamental object, along with the string.
String theory has 9 space dimensions, so a brane can have anywhere from 0 to 9 dimensions. Branes were hypothesized as part of string theory in the late 1980s. In 1995, Joe Polchinski realized that Edward Witten's proposed M-Theory required the existence of branes.

Some physicists have proposed that our own universe is in fact a 3-dimensional brane, on which we are "stuck" within a larger 9-dimensional space, to explain why we can't perceive the extra dimensions.

Theory of Superstring, and M Theory

The followings provide a passing glance on the parts of Superstring theory leading to the adoption of the Calabi-Yau space.

Extra Dimensions and Supersymmetry - In the process of developing the Superstring theory, a number of problems arose such that the theory makes sense only if there are extra-dimensions beyond the usual 3 spatial and 1 temporal dimensions or by including supersymmetry into the theory. The problems can be summarized as :

Ghost - It usually means negative probability. This is associated with the failure to construct physical states from the time component in the mode expansion (of the position vector Xµ ) and the subsequent quantization. Such problem always comes up with the bosonic string. The solution is to impose the condition that the dimension D = 26. Then 16 of these 26 extra-dimensions are compactified into small circles leaving 6 more dimensions to be compactified further from the remaining 10 dimensions. ... The d11 dimension is the additional dimension demanded by the M Theory. The blue line is the 3 brane in which all of us live in.
Anomaly - An anomaly is the failure of a classical symmetry (such as Lorentz invariance) to survive the process of quantization or regularization (a mathematical device to bypass undesirable expression such as infinity). Conservation laws are violated as the consequence. For the bosonic string the solution is to impose the same restriction for the dimension D = 26, while in the superstring theory with both bosonic and fermionic degrees of freedom, the condition becomes D = 10 (after compactifying 16 bosonic dimensions in the Heterotic theories) .
Inconsistency - This is the appearance of infinities in the theory. The same solution of taking D = 26 can be applied to the bosonic string, and D = 10 to the superstring.
Tachyon - In the bosonic string theory, tachyon appears in its formulation. Tachyon is the kind of particle always moving faster than the speed of light. It has never been observed, and it creates instability in the theory - decaying to lower energy state by unknown mechanism. This troublesome feature is removed by incorporating supersymmetry into the theory.
Supersymmetry - In addition, introduction of supersymmetry into the string theory fixes the problems of missing fermions, offers a way to produce realistic mass for the particles, stabilize the vacuum, drops the dimensional requirement from 26 to 10, and generally helps to reduce the labor of calculations.

Compactification - Since all of us experience only 3 spatial and 1 temporal dimensions, the 10 and 26 extra-dimensions have to be hidden under some schemes. One of the two alternatives is to roll them up into very small size not observable even under a very powerful microscope. The other one is to consider our existence on a 3 brane floating in the bulk of ten spatial dimensions. The first alternative is called compactification

The Unification of Gravity and Quantum Mechanics

Theorists have long sought a way to obtain a quantum-mechanical description of gravity, the one force that has resisted integration into the quantum framework. In this, one of string theory's most compelling strengths lies. It has the ability to integrate gravity into its own framework, which also includes quantum mechanics, thus providing physicists with the first such theory in history. The difficulty, as stated in Introduction to String Theory, is the fact that the uncertainty principle allows "virtual" particles to "borrow" energy in wild fluctuations provided that they relinquish it within a certain amount of time. These quantum fluctuations become ever more evident as they are examined on smaller and smaller scales, finally leading to the roiling quantum foam of sub-Planck-length distances. This contrasts sharply with the smooth geometrical surfaces postulated and required by relativity.

Strings vs. Point Particles

The main distinction between strings and point particles is the fact that strings have spatial extent, whereas point particles are literally zero-dimensional. It had previously been supposed by physicists formulating quantum mechanics that the elementary particles were points. However, string theory states that the elementary particles are not points but instead are tiny undulating strings. According to the uncertainty principle, the ability of a particle to "probe" an area depends on its quantum wavelength, or the amount of uncertainty in its position - in other words, a particle's sensitivity becomes "blurred" by quantum jitters. As you will see, this proves to be important in the unification of quantum mechanics and string theory.

Point Particles as Probes

A particle's quantum wavelength, or its "jitteriness," is inversely proportional to its momentum, which for the purposes of this discussion can represent its energy. Therefore, if a particle's energy is increased, its quantum wavelength decreases - it can be made clearer and clearer - and it can be used to probe smaller and finer objects. According to this formulation, there is no limit to the energy with which a particle can be imbued, so there is also no limit to the smallness of the objects it can probe. This is what gives rise to the quantum foam on sub-Planckian distance scales.

Strings as Probes

This is where the differences between strings and point particles become most evident. Strings, which are about the Planck length, cannot probe distances smaller that themselves - meaning they cannot even reach sub-Planckian distances. Furthermore, increasing the energy of a string does not necessarily increase its probing ability by decreasing its quantum wavelength. Up to certain energies, added energy will decrease this blurring, but when energy beyond that required to probe Planck-length distances, energy actually causes strings to grow. This makes it less sensitive a probe of short distances. In fact, the energy attained during the Big Bang could theoretically have made a string grow to macroscopic size. This means that a string actually has two sources of the blurring discussed earlier - quantum jitters and its own size. Since adding energy decreases blurring from the first, but eventually increases it from the second, one cannot probe sub-Planckian distances if strings are the fundamental units of the universe - if the smallest unit of the universal hierarchy cannot probe these distances, then quantum foam cannot ultimately affect anything except in a few special conditions. In fact, if something must be measurable in order to exist, it can be said that this quantum foam does not actually exist. It is simply a mathematical artifact arising out of an imprecisely formulated theory.

String theory is unfalsifiable. Because it's not a theory. It's not a hypothesis. It is metaphysical. It is the language from which all physics is derived.