What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work? Almost any mathematical concept has antecedents; it builds on, or is related to, previously known concepts. But are there concepts that owe little or nothing to previous work?
The only example I know is Cantor's theory of sets. Nothing like his concrete manipulations of actual infinite objects had been done before.
 A: Category theory must be here too - although it was created not so much as something out of the blue but rather to organise and interrelate the accumulated body of mathematical knowledge (according to Eilenberg and MacLane categories were invented to formulate rigorously the intuitive notion of natural transformation), still I think it was a completely new approach to the very idea of abstraction in mathematics which I believe has yet to show us its full potential.
A: Although his work was certainly related to earlier fields, I believe that Ramanujan (pretty much) built up a lot of his work from scratch.  
A: Graph theory is an example that comes to mind, via the problem of the seven bridges of Königsberg : http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg . 
A: Écalle's work on resummation and resurgent functions.  While there is a bit of work that pre-dates him, the vast bulk of his theory is really novel and built 'from scratch'.  This is especially clear to anyone who has ever tried to read the Orsay preprints of his original manuscripts on resurgent functions!  [The only notation more spectacular than his was Frege's]
A: Pcf theory/cardinal arithmetic. Well, it's not exactly built from scratch, but there are plenty of nice results which do not use any sophisticated metamathematical machinery (such as forcing, inner models, etc).
Edit: I've deleted part of my answer due to a little misunderstanding.
A: Stallings's bipolar structures created to prove that groups of cohomological dimension 1 are free.
(Stallings might not have agreed with my nomination, but his statement at the end of the paper that his techniques are a result of "meditating on the proof of the Sphere Theorem" somehow makes his work even more remarkable to me.)
A: Shannon's work on Information theory. Maybe the math wasn't new but the ideas (such as positing a qualitative metric of information and identifying its relevance to design of communication systems) definitely were.
A: Hermann Weyl wrote in a 1939 article on invariant theory: "The Theory
of Invariants came into existence about the middle of the nineteenth century
somewhat like Minerva: a grown-up virgin, mailed in the shining armor of
algebra, she sprang forth from Cayley's Jovian head. Her Athens over which
she ruled and which she served as a tutelary and beneficent goddess was
projective geometry."
A: There is another example in Set Theory, which is Paul Cohen's forcing. Of course, forcing had some ties with earlier work, but the bulk of it was completely new.
A: I wonder if Euler deducing the infinitude of primes from the divergence of the harmonic series or Riemann's work on the Riemann zeta function would be suitable examples?
A: Random Graphs.
Started by Paul Erdos and Alfred Renyi.
A: J. Von Neumann introduced the concept of a continuous geometry over a division ring in 1936. 
In these geometries that are extensions of the well known finite dimensional projective geometries, there is a dimension function that takes all values in the interval [0,1]. I guess these are the first examples of "pointless" geometries, that is geometries that are not made of points or atoms (i.e. there are no minimal elements for the order that corresponds to the inclusion in the usual projective geometries).
A: I have to agree with Scott's comment: Every development has its roots. The following three examples are thus only approximations. 
The first is Riemann's work on the "On the Hypotheses which lie at the Bases of Geometry". As a habilitation talk it is almost devoid of any details, but it is not only one of the earliest accounts of geometry in $n$ (or even infinite) dimension, it also gives the ideas of a Riemannian metric and the Riemann curvature tensor! As Riemann said it:

[...] ausser einigen ganz kurzen Andeutungen, welche Herr Geheimer Hofrath
  Gauss in der zweiten Abhandlung über die biquadratischen Reste [...] darüber gegeben hat und einigen philosophischen Untersuchungen Herbart’s, [konnte ich] durchaus
  keine Vorarbeiten benutzen [...].
Translation: expect for a few very short hints, which Privy Councillor Gauss gave in his second work on biquadratic residues, and some philosophical investigations Herbart's, I could not use any previous work. 

Also Gauss's work on the relationship between intrinsic and extrinsic geometry of surfaces, culminating in his Theorema Egregium, might qualify. Of course, there was some previous work on surfaces, but this goes so much deeper that all previous work pales in comparison. 
I also want to mention Grassmann's Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, which already states in the title that is a new branch of mathematics. (Note there are two quite different editions, 1844 and 1862). Essentially he invented linear algebra in this book. Again not completely without precursors, as people solved linear equations before, but to use geometric ideas in $n$ dimension, subspaces, linear independence, exterior algebras etc. was very new. See this this article for an overview of his contributions.
A: The solution of the cubic equation by Scipione del Ferro and Tartaglia
in the early 16th century. This was not only a great advance in algebra,
but it also forced mathematicians to confront complex numbers.
A: This happened hundreds of times in physics throughout the twentieth century, because physicists were specifically trained to do mathematics from scratch. The main reason is that it was too time consuming in pre-internet times to learn the specialized jargon of each subfield, so it was easier just to rederive the stuff.
The most significant early success of this sort of willful ignorance is probably the development of special relativity from essentially nothing. The Minkowski geometry of relativity is remarkable, because if you interpret the words "point" and "line" as usual, and the word "circle" as a unit hyperbola with 45-degree angle asymptotes (the unit circle of relativity),  it satisfies all the explicit axioms of Euclid's geometry, as set out in the elements, including the axiom of parallels, but is not Euclidean. The essential difference is that circles are not closed curves, so that certain implicit betweenness properties fail. There are distinct points which are at a zero "distance" from one another, the hypotenuse of a right triangle is always shorter than one of the sides, etc. This is amazing to me, because of the number of people who have considered models of geometry before Einstein (including all the heavy focus on non-Euclidean geometry for the previous century). All the bigwigs missed Minkowski geometry.
Aside from Einstein's work, there are the following mathematical developments from physics, all of which came out of nowhere mathematically:


*

*Quantum mechanics, in particular, the theory of the canonical commutation relation [x,p]=i and its relationship with wave operators and random walks.

*Dirac's distribution theory (delta-functions): this completed the notion of Eigenvalue of a linear operator to include Eigenvalues and Eigenfunctions for the x operator in quantum mechanics.

*Majorana spinors--- these were due to the discovery of the Dirac equation. The representation theory of SO(p,q) is now entirely dependent on dirac matrices and the Majorana and Weyl conditions.

*Wigner's random matrix theory. This was completely ab-initio, and is now very active mathematics.

*Anderson localization: this is also a mathematical surprise--- the eigenfunctions of randomized potentials are localized in space. The full resulting theory has still not been made part of mathematics, but Anderson's paper is an ab-initio (although not rigorous) argument.

*Metropolis algorithm--- this essentially inaugurated monte-carlo methods, and I do not know any previous work it builds on.

*Feynman's path integral--- this was developed within mathematics as the Wiener integral at about the same time, but the physics work is completely ab-initio. Needless to say, the results are not going into mathematics easily (in my opinion, this is mostly due to the reluctance of mathematicians to make every subset of R measurable).

*Candlin's fermionic path integral (Berezin integrals)--- Candlin in 1956 develops the whole theory of path integrals for fermionic fields from scratch in a Neuvo Cimento article with next to no citations (in either direction). The theory was ignored for a decade for no apparent reason.

*Mandelstam's double dispersion relations (and dispersion relations in general).

*Kraichnan's inverse cascade--- generally the statistical theory of nonlinear classical equations is developed from scratch by Kraichnan and others. The biggest shocker is the inverse cascade--- in two dimensions, eddies go up from small scales to big scales.

*Zimmermann's forest formula--- this is now part of mathematics, due to Kreimer and Connes, but Zimmermann did it from scratch in physics.

*The theory of second order phase transitions and modern renormalization by Widom/Wilson.

*Wilson's theory of operator product expansions, (which is not a part of mathematics yet)

*Supersymmetry is developed from scratch by several groups with no previous motivation in mathematics (not much in physics). The original germ of an idea is in Golfond and Likhtman, but the person who does most of the early theory work is Pierre Ramond. Wess and Zumino's work also comes out of nowhere.

*Virasoro algebra/Kac-Moody algebra-- Virasoro algbera is the theory of infinitesimal conformal maps under composition, so it should have been classical mathematics, but as far as I know, it wasn't. The theory started (as far as I know) with the study of string theory in the early 1970s.

*Mirror symmetry--- this owes to previous work in T-duality in string theory, not in mathematics. 

*Witten's global anomalies--- these are not yet part of rigorous mathematics, but they are ab-initio, and were a complete surprise.


I got tired, but there are hundreds, maybe thousands of examples, because all the results in the physics literature were generally ab-initio. It is a standard practice for some mathematicians to scan the physics literature for original ideas and incorporate them into mathematics.
A: The existence of irrational numbers.
A: This is an intruiging question. I have some suggestions but I am not sure about them.
1) Frege's work on logic. (Logic was stagnated for many many centuries before.)
2) Conway's surreal numbers.
3) Game theory (e.g. zero sum games).
A: It seems like Dirichlet's Theorem on Primes in Arithmetic Progressions came out of nowhere, or at least his methods of proof.  While the complex analysis may not have been new, his application of it, through the Dirichlet characters and the series he made from them, to number theory was pretty novel.
A: The Analytic Geometry of Rene Descartes.
A: Cauchy's development of the theory of complex integration and then Riemann's extension of this to surfaces.
A: laplace transforms as initiated by euler
