A chart is a continuous isomorphism with continuous inverse between a neighbourhood in a topological space (e.g. a smooth manifold) and a neighbourhood in some vector space.
Two charts whose neighbourhoods in the topological space overlap may be used to define a continuous isomorphism between two neighbourhoods in the the vector space: these are the images of the overlap under the two charts (or their inverses, depending on which way round you defined the charts) and the isomorphism is the composite of one chart's restriction and the other's inverse's restriction to relevant neighbourhoods. As this isomorphism is now between two portions of a vector space, we may ask whether it is smooth (differentiation being readily enough defined on a vector space). If it is, we shall describe the two charts as smoothly commensurate.
An atlas is a collection of charts whose neighbourhoods in the topological space form a cover of that space, i.e., their union is the entire space, and whose vector spaces are all the same.
An atlas is described as smooth if all overlapping pairs of its charts are smoothly commensurate on their overlaps.
It is worthy of note that we can define the completion of a smooth atlas to be the collection of all charts, using the same vector space, smoothly commensurate with every chart of the atlas. This is known as the completion of the original atlas.
The usual definition of a smooth manifold is that it is a topological space equipped with a smooth atlas. It is said to have the dimension of the vector space used in the charts of the atlas.
Any morphism to or from a neighbourhood in a smooth manifold is then said to be smooth if its composite with any chart composable with it (restricting each to a subset of its domain as appropriate) is smooth. For a morphism between neighbourhoods in the smooth manifold, for instance, this means that going from the vector space, along a chart, along the morphism and then back to the vector space via another chart, wherever this is possible, must be smooth morphism from the vector space to itself.
It is worthy of note that the construction I give for the tangent and gradient bundles of a smooth manifold may, just as readily, be used to define a smooth manifold. To see the equivalence of the two definitions, note that a family of scalar fields on some open neighbourhood of the manifold whose derivatives form a basis of the gradient bundle throughout that neighbourhood constitute a chart. The interconvertibility of bases then suffices to deliver the smooth equivalence of any two charts on their overlap.
Written by Eddy.