A unitary representation is defined as a representation where all generators $G$ act as hermitian matrices: $$ G^\dagger G= 1$$
To make sense of the probabilistic interpretation of quantum theory, we need to use unitary representations (see e.g. page 69 in Schwartz's book). Expressed differently, we are interested in representations of given groups on the Hilbert space in a quantum field theory.
Many important groups are non-compact (e.g. the Poincare group and the conformal group) and there is a theorem that tells us that all unitary representations of a non-compact group are infinite-dimensional.
“The infinite-dimensional representations are considered unphysical because we never see particle states in nature labelled by extra continuous parameters.” http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy2001.pdf
Thus it is often a non-trivial problem to find make sense of the unitary representations of such groups.