We consider a class of states in an ensemble of two-level atoms: a mesoscopic superposition of two distinct atomic coherent states, which can be regarded as atomic analogs of the states usually called Schrödinger-cat states in quantum optics. According to the relation of the constituents, we define polar and nonpolar cat states. The properties of these are investigated by the aid of the spherical Wigner function. We show that nonpolar cat states generally exhibit squeezing, the measure of which depends on the separation of the components of the cat, and also on the number of the constituent atoms. By solving the master equation for the polar cat state embedded in an external environment, we determine the characteristic times of decoherence and dissipation, and also the characteristic time of a new parameter, the nonclassicality of the state. This latter one is introduced by the help of the Wigner function, which is used also to visualize the process. The dependence of the characteristic times on the number of atoms of the cat and on the temperature of the environment shows that the decoherence of polar cat states is surprisingly slow.