"A differentiable manifold is a topological space on which we can do calculus", is the short definition. The long definition calls for a series of subdefinitions. For best results, take small bites and chew slowly and thoroughly.
Let
M be a
Hausdorff topological space.
A chart φ is a homeomorphism from some open subset U of M to an open subset φ(U) of some Euclidean space R^{n}.
An atlas on A is a collection of open sets {U_{i}  i ∈ I} that cover M together with charts φ_{i} : U_{i} > R^{n} that satisfy the following compatibility condition: for any pair of charts (U_{i}, φ_{i}), (U_{j}, φ_{j}) both of the transit maps
φ_{i}φ_{j}^{1} : φ_{j}(U_{i} ∩ U_{j}) > φ_{i}(U_{i} ∩ U_{j})
and
φ_{j}φ_{i}^{1} : φ_{i}(U_{i} ∩ U_{j}) > φ_{j}(U_{i} ∩ U_{j})
are homeomorphisms between open subsets of Euclidean space. In other words, wherever two charts give coordinates for the same portion of a manifold, there is a continuous changeofcoordinate map from one chart to the other with a continuous inverse. Notice that by the highly nontrivial Invariance of Domain theorem, this compatibility condition implies that any chart between M and Euclidean space must in fact be to Euclidean nspace, as there is no homeomorphism between Euclidean spaces of different dimensions. If M is connected, it therefore makes sense to speak of the dimension n, given by any chart, as all charts map to a Euclidean space of the same dimension.
Have you digested those definitions? Did you draw all the appropriate commutative diagrams that show what maps to what? Good! Then you're ready to understand what a manifold is.
A manifold is a Hausdorff topological space together with an atlas.
Requiring the transit maps to be homeomorphisms is not a very strong condition. It is the minimal condition necessary so that our intuitions about Euclidean space can extend to the manifold. Since the transit maps are homeomorphisms between Euclidean space (from R^{n} to R^{n}, say), we have further structure we can impose upon these maps. Namely, we can speak about the (Frechet) derivative between two real normed vector spaces (read "derivative" as "best linear approximation"). This calls for further definitions.
A differentiable structure is an atlas with the additional conditions that

The transit maps are of class C^{r} or class C^{∞}. This means, respectively, that the transit maps are either rtimes continuously differentiable or infinitely differentiable. For convenience and to avoid annoying questions that belong to analysis and not geometry, we usually never bother with C^{r} at all and only focus on C^{∞} transit maps.

The atlas is maximal. This means that if you can come up with another chart that is compatible with every other chart already in the atlas, then the chart you came up with was already in the atlas to begin with. A more formal statement of maximality is left to the diligent reader.
It should not be too surprising to hear now that a differentiable manifold is a Hausdorff topological space together with a differentiable structure. Sometimes, when speaking of differentiable manifolds and general manifolds without a differentiable structure, the latter are referred to as topological manifolds in order to emphasize that no assumption on differentiability is made.
A natural question that arises upon first working through these definitions is whether a manifold can admit more than one differentiable structure. "Surely not!" is our naïve first guess. We cite as evidence the facts that spheres of dimension 1,2,3,5 and 6 (not four!) only admit one differentiable structure. It thus came as a great surprise when Milnor came up with TWENTYEIGHT nonequivalent differentiable structures for the sevensphere. The number of inequivalent differentiable structures for the next few spheres are 2, 8, 6, and, of all wackedup numbers out there, 992. Questions about this sort of phenomena belong to the branch of mathematics known as differential topology.