the state space
If the index set says when you look, the state space says what you can see. It is the set of all possible values each random variable X_t is allowed to take — the menu of states the process can be in at any moment.
Like the index set, it splits into discrete and continuous. A discrete state space is a separated set of values: the integers (a random walk's position), or a small named list like {sunny, cloudy, rainy} or {working, broken}. A continuous state space is a continuum: the real line (a price), an interval, or a region of the plane (a particle's location). The two choices — discrete or continuous time, discrete or continuous state — combine into a useful 2-by-2 grid. A Markov chain is typically discrete-time, discrete-state; the Poisson count is continuous-time, discrete-state (it jumps by integers but at real-valued moments); Brownian motion is continuous-time, continuous-state.
The state space governs the toolkit. Discrete states let you use a transition matrix — a table of probabilities of going from each state to each other state — and arrange everything as vectors and matrices. Continuous states force you to use densities and integrals instead, since you cannot list the states one by one. Knowing the state space and the index set immediately tells you which corner of stochastic-process theory you are standing in.
A weather model with X_t taking values in {sunny, rainy} has a discrete state space of just two states. A stock-price model with X_t taking any value in (0, infinity) has a continuous state space. The simple random walk on the line has state space {..., -2, -1, 0, 1, 2, ...}, the integers — discrete state, discrete time.
Time index (when) and state space (what) together place a process in the 2-by-2 grid of process types.
A 'discrete-state' process can still live in continuous time and vice versa — do not confuse the two axes. The Poisson process is the classic trap: its values are integers (discrete state) yet its jumps occur at continuous-time instants.