Rn is n dimensional space, the product of n copies of the real line. Note that Rn is more than just a set of points, each identified by n coordinates; Rn is a metric space, with an implied topology. The metric is the euclidean distance metric, although other metrics will produce the same topology.
Cn is n dimensional complex space, using the associated metric. If you're not multiplying points together (via complex multiplication), Cn is indistinguishable from R2n. The points are the same; the metric is the same; the topology is the same.
The set of vectors in Rn+1 with norm 1 is denoted Sn, the n-sphere. The circle in the plane is denoted S1, and the sphere in 3 space is denoted S2. The vectors in Rn that have norm ≤ 1 form Dn, the n dimensional disk. Thus the boundary of D2 is S1.
The direct product of S1 n times gives the n-torus, or Tn. T2 is the traditional torus, like an inner tube. One circle runs around the outside of the tube and the other defines the cross section of the tube. The product of these two circles defines the entire torus, with the product topology.
Take the nonzero points of Rn+1 and identify a point with all its scalar multiples to get real projective space, denoted RPn, or simply Pn. This is equivalent to identifying antipodal points on Sn. Similarly, identify scalar multiples in Cn+1 to get complex projective space, CPn.