For time series without seasonality, the most basic form is the linear trend + noise model: $X_{t}=\alpha + \beta t +\epsilon_{t}$Such a trend with a deterministic form is a **global** trend. To allow more flexibility, a **stochastic** trend allows the coefficients $\alpha$ and $\beta$ to change stochastically over time.
More generally, a trend can be non-linear, e.g. quadratic and exponential growth.
The analysis of trends depends on the goal: whether it is to measure the trend itself, or to remove the trend for analyzing local patterns.