A unit root is a property of a non-stationary time series which can lead to spurious regressions and wrong inference. A series $$y_t = y_{t-1} + e_t$$ with $e_t \sim iid(0,\sigma^2)$ has a unit root if it can be expressed as $(1-L)y_t = e_t$ where $L$ is the lag operator. Then the characteristic equation of the above process has one unit root. A property of the unit root is that when an $I(1)$ autoregressive process is first differenced, it becomes an $I(0)$ process, i.e. it will be stationary.
Alternatively we can write: $$y_t = \phi y_{t-1} + e_t$$ The standard case is $\phi = 1$ in which shocks to the system ($e_t$) are persistent. For $\phi<1$ they die out over time and for $\phi>1$ we have an explosive unit root. The latter case is usually ignored since this rarely happens in reality.