Wikipedia-style explanation of Geometric Complexity Theory
I'm not exactly sure what level is suitable for Wikipedia article (different articles seem to be aimed at different levels of expertise) or exactly what you're looking for. So here's a try, but I'm open to feedback.

Geometric complexity theory proposes to study the computational complexity of computing functions (say, polynomials) by exploiting the inherent symmetries in complexity and any additional symmetries of the functions being studied. 

As with many previous approaches, the ultimate goal is to separate two complexity classes $\mathcal{C}_{easy}, \mathcal{C}_{hard}$ by showing that there is a polynomial $p$ which takes functions $f$ as inputs (say, by their coefficient vectors) such that $p$ vanishes on every function $f \in \mathcal{C}_{easy}$ but does not vanish on some function $g_{hard} \in \mathcal{C}_{hard}$. 

The first key idea (cf. [GCT1, GCT2]) is to use symmetries to organize not the functions themselves, but to organize the (algebro-geometric) properties of these functions, as captured by polynomials such as $p$ above. This enables the use of representation theory in attempting to find such a $p$. Similar ideas relating representation theory and algebraic geometry had been used in algebraic geometry before, but to my knowledge never quite in this way.

The second key idea (cf. [GCT6]) is to find combinatorial (and polynomial-time) algorithms for the resulting representation-theoretic problems, and then reverse-engineer these algorithms to show that such a $p$ exists. This may be taken in the spirit of using Linear Programming (an algorithm) to prove certain purely combinatorial statements. 

Indeed, [GCT6] suggests reducing the representation-theoretic problems above to Integer Programming problems, then showing that the resulting IPs are solved by their LP relaxations, and finally giving combinatorial algorithms for the resulting LPs. The conjectures in [GCT6] are themselves motivated by reverse-engineering known results for the Littlewood-Richardson coefficients, an analogous but easier problem in representation theory. In the case of LR coefficients, the Littlewood-Richardson combinatorial rule came first. Later Berenstein and Zelevinsky [BZ] and Knutson and Tao [KT] (see [KT2] for a friendly overview) gave an IP for LR coefficients. Knutson and Tao also proved the saturation conjecture, which implies that the IP is solved by its LP relaxation (cf. [GCT3,BI]).

The results of [GCT5] show that explicitly derandomizing Noether's Normalization Lemma is essentially equivalent to the notorious open problem in complexity theory of black-box derandomization of polynomial identity testing. Roughly how this fits into the larger program is that finding an explicit basis for the functions $p$ that (do not) vanish on $\mathcal{C}_{easy}$ (in this case, the class for which the determinant is complete) could be used to derive a combinatorial rule for the desired problem in representation theory, as has happened in other settings in algebraic geometry. An intermediate step here would be to find a basis for those $p$ that (do not) vanish on the normalization of $\mathcal{C}_{easy}$, which is by construction a nicer algebraic variety -- in other words, to derandomize Noether's Normalization Lemma for DET.

Examples of symmetries of complexity and functions

For example, the complexity of a function $f(x_1, \dotsc, x_n)$ - for most natural notions of complexity - is unchanged if we permute the variables $f(x_{\pi(1)}, \dotsc, x_{\pi(n)})$ by some permutation $\pi$. Thus permutations are symmetries of complexity itself. For some notions of complexity (such as in algebraic circuit complexity) all invertible linear changes of the variables are symmetries.

Individual functions may have additional symmetries. For example, the determinant $\det(X)$ has the symmetries $\det(AXB) = \det(X^{T}) = \det(X)$ for all matrices $A,B$ such that $\det(AB) = 1$. (From what little I picked up about this, I gather that this is analogous to the phenomenon of spontaneous symmetry-breaking in physics.) 

Some Recent Progress [this section definitely incomplete and more technical, but a complete account would take tens of pages....I just wanted to highlight some recent progress]

Burgisser and Ikenmeyer [BI2] showed a $\frac{3}{2}n^2$ lower bound on matrix multiplication following the GCT program as far as using representations with zero vs nonzero multiplicities. Landsberg and Ottaviani [LO] gave the best known lower bound of essentially $2n^2$ on the border rank of matrix multiplication using representation theory to organize algebraic properties, but not using representation multiplicities nor combinatorial rules. 

The next problem after Littlewood-Richardson coefficients is the Kronecker coefficients. These show up both in a series of problems that is suspected to eventually reach the representation-theoretic problems arising in GCT, and more directly as bounds on the multiplicities in the GCT approach to matrix multiplication and permanent versus determinant. Finding a combinatorial rule for Kronecker coefficients is a long-standing open problem in representation theory; Blasiak [B] recently gave such a combinatorial rule for Kronecker coefficients with one hook shape.

Kumar [K] showed that certain representations appear in the coordinate ring of the determinant with nonzero multiplicity, assuming the column Latin square conjecture (cf. Huang-Rota and Alon-Tarsi; this conjecture also, perhaps not coincidentally, shows up in [BI2]). Hence these representations cannot be used to separate permanent from determinant on the basis of zero vs nonzero multiplicities, though it still might be possible to use them to separate permanent from determinant by a more general inequality between multiplicities.

References
[B] J. Blasiak. Kronecker coefficients for one hook shape. arXiv:1209.2018, 2012.

[BI] P. Burgisser and C. Ikenmeyer. A max-flow algorithm for positivity of Littlewood-Richardson coefficients. FPSAC 2009.

[BI2] P. Burgisser and C. Ikenmeyer. Explicit Lower Bounds via Geometric Complexity Theory. arXiv:1210.8368, 2012.

[BZ] A. D. Berenstein and A. V. Zelevinsky. Triple multiplicities for $\mathfrak{sl}(r+1)$ and the spectrum of the exterior algebra of the adjoint representation. J. Algebraic Combin. 1 (1992), no. 1, 7–22. 

[GCT1] K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems. SIAM J. Comput. 31(2), 496–526, 2001.

[GCT2] K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties. SIAM J. Comput., 38(3), 1175–1206, 2008.

[GCT3] K. D. Mulmuley, H. Narayanan, and M. Sohoni. Geometric complexity theory III: on deciding nonvanishing of a Littlewood-Richardson coefficient. J. Algebraic Combin. 36 (2012), no. 1, 103–110. 

[GCT5] K. D. Mulmuley. Geometric Complexity Theory V: Equivalence between blackbox derandomization of polynomial identity testing and derandomization of Noether's Normalization Lemma. FOCS 2012, also arXiv:1209.5993. 

[GCT6] K. D. Mulmuley. Geometric Complexity Theory VI: the flip via positivity., Technical Report, Computer Science department, The University of Chicago, January 2011. 

[K] S. Kumar. A Study of the representations supported by the orbit closure of the determinant. arXiv:1109.5996, 2011.

[LO] J. M. Landsberg and G. Ottaviani. New lower bounds for the border rank of matrix multiplication. arXiv:1112.6007, 2011.

[KT] A. Knutson and T. Tao. The honeycomb model of $\text{GL}_n(\mathbb{C})$ tensor products. I. Proof of the saturation conjecture. J. Amer. Math. Soc. 12 (1999), no. 4, 1055–1090. 

[KT2] A. Knutson and T. Tao. Honeycombs and sums of Hermitian matrices. Notices Amer. Math. Soc. 48 (2001), no. 2, 175–186.