a chart and atlas
When you open a road atlas of a country, no single page shows the whole land without distortion; instead each page shows one region flattened onto the paper, and adjacent pages overlap so you can trace a road from one to the next. A chart of a manifold is exactly one such page: a faithful flattening of a small piece of a curved space onto ordinary coordinate space. An atlas is the whole book — a collection of charts that together cover every point, with sensible overlaps.
Precisely, a chart on a manifold M is a pair: an open region U of M, together with a map phi that sends each point of U to a tuple of numbers (x^1, ..., x^n) in R^n, in a one-to-one, continuous, continuously-invertible way. Those numbers are the local coordinates of the point. An atlas is a family of charts whose regions cover all of M. Where two charts overlap, going from one chart's coordinates to the other's is done by a transition map; the manifold is called smooth precisely when every transition map is infinitely differentiable. The transition maps are where all the gluing information lives — they tell you how the local pictures fit together into one consistent space.
Charts are the indispensable tool for computation: to differentiate a function on a manifold, you pull it back through a chart into honest R^n and use ordinary calculus. But a chart is a choice, not a feature of the space — like choosing latitude-longitude versus a city grid. Any quantity that genuinely belongs to the geometry must come out the same no matter which chart you compute it in; this 'independence of coordinates' is the central discipline of the whole subject. A coordinate singularity, such as the way longitude becomes meaningless at the poles, is a flaw of the chart, not of the manifold.
On Earth, latitude-longitude is one chart. It works beautifully over most of the globe but breaks at the poles, where every longitude meets and 'east' loses meaning. This is a coordinate singularity, not a hole in the planet — switching to a different chart near the pole (say a small flat grid) restores good coordinates there.
Longitude failing at the poles is a fault of the chart, not the sphere — the cure is to overlap a second chart.
Beginners often treat coordinates as if they were the space itself. They are not: coordinates are a removable scaffold, and a real geometric statement must be checkable in any overlapping chart and give the same answer.