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A unit root (also called a unit root process or a difference stationary process) is a stochastic trend in a time series, sometimes called a “random walk with drift”; If a time series has a unit root, it shows a systematic pattern that is unpredictable.
The reason why it’s called a unit root is because of the mathematics behind the process. At a basic level, a process can be written as a series of monomials (expressions with a single term). Each monomial corresponds to a root. If one of these roots is equal to 1, then that’s a unit root.
The math behind unit roots is beyond the scope of this site (although if you’re really interested you can read about it in this pdf). All you really need to know if you’re analyzing time series is that the existence of unit roots can cause your analysis to have serious issues like:
Spurious regressions: you could get high r-squared values even if the data is uncorrelated.
Errant behavior due to assumptions for analysis not being valid. For example, t-ratios will not follow a t-distribution.
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