Why are they so important?
I see them appear in Lie theory, algebraic geometry, etc.
Can somebody elaborate?
For example, can someone explain why they are such natural groups to consider in algebraic geometry?
They are solvable groups and there representation theory is by far easier than the general representation theory (Mackey-Clifford-etc.)
In principle, it is sufficient to classify the representations of unipotent groups and reductive groups, in order to classify all the representation in full.
Every lie or algebraic group is a semi direct product of a reductive one and a unipotent one.
Also the reductive subgroups of a reductive group determine to a large extent the representation theory via parabolic induction. This involves necessary understanding the parabolic subgroups, hence the unipotent subgroups.