When $c=4$, the map $T_4(x)=4x(1-x)$ on the unit interval is semi-conjugate to the transformation $z\mapsto z^2$ of the unit circle via $z\mapsto\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)$: $$ T_4\left(\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)\right)=\frac{1}{2}-\frac{1}{4}\left(z^2+\frac{1}{z^2}\right). $$ Thus the pushforward of the Lebesgue measure on the unit circle (which is an ergodic measure for $z\mapsto z^2$) by $z\mapsto\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)$ provides an an absolutely continuous ergodic measure for $T_4:[0,1]\rightarrow [0,1]$.
In general, there is a very deep theorem of Lyubich stating that, aside from a measure zero subset of parameters, for every $c$ the map $T_c$ is either hyperbolic (i.e. has an attracting periodic orbit) or stochastic (i.e. admits an absolutely continuous invariant measure).
Added: Another well-studied example is $c\approx 3.57$ that happens at the end of period-doubling cascade $-$ the largest parameter for which the topological entropy is zero. For this parameter, the map $T_c$ is infinitely renormalizable (the aforementioned paper of Lyubich shows that the set of such parameters is of measure zero). For this $c$, there exists a Feigenbaum attractor. This is an invariant Cantor set on which the dynamics of $T_c$ is conjugate to a "2-adic adding machine". Ergodic probability measures for such an interval map are classified here; these are supported either on periodic orbits of period $2^n$ or on the Feigenbaum attractor.