Perhaps a simple example from finance might help intuition. Let $R_t$ be the interest rate for period $t$ (note this is a random variable).
The interest rate is typically modeled as a stationary process. If you earn the interest rate $R_t$ each period and start with $V_0$ dollars, then the quantity of dollars you have at time $t$ is given by:
$$V_t = V_0 \prod_{\tau=1}^t \left(1 + R_\tau \right)$$
The process $\left\{ V_t \right\}$ is NOT stationary. There's no unconditional mean or variance. On the other hand, the interest rate $R_t$ is typically modeled as a stationary process. An extremely simple model for the interest rate might be $R_t = a + b R_{t-1} + \epsilon_t$. (In fact, the Vasicek model is the continuous time analogue to that AR(1) model of the interest rate.)
Other examples from econ and finance:
Let $Y_t$ be aggregate output (i.e. GDP) of the economy at time $t$.
- $y_t = \log(Y_t)$ is almost certainly not a stationary process.
- The growth in log output (i.e. $y_t - y_{t-1}$) is typically treated as a stationary process
Let $S_t$ be the price of overall market portfolio.
- $s_t = \log(S_t)$ is almost certainly not a stationary process.
- The log return $r_t = s_t - s_{t-1}$ of the market portfolio is typically treated as a stationary process.
A random walk or a Wiener process (the continuous time analogue to a random walk) are canonical examples of non-stationary processes. On the other hand, increments of a random walk or a Wiener process are stationary processes.
Temperature
As @kjetil points out, temperature is not a stationary process. For example, the distribution over temperatures in January is not the same as the distribution over temperatures in June. The joint distribution changes when shifted in time.
On the other hand, let $\mathbf{y}_t$ be a 12 by 1 vector for year $t$ where each entry of the vector denotes the average temperature for a month. You might be able to argue that $\mathbf{y}_t$ is a stationary process?
Sunspots
One of the first time-series models was developed by Yule and Walker to model the 11-year sunspot cycle.
Let $y_t$ be the number of sunspots in year $t$. They modeled the number of sunspots in a year as a stationary process using the AR(2) model:
$$ y_t = a + b y_{t-1} + c y_{t-2} + \epsilon_t $$
A stationary process can have patterns, cycles, etc...