The answers given by Bruce and Anton are sensible, but I'd add that it's important in Lie theory to distinguish the behavior of reductive groups from that of arbitrary Lie groups: finite dimensional representations of the former are semisimple (motivating the label "reductive") but often in general. It's also important to distinguish finite and infinite dimensional representations, since you use the term "continuous". Even for a well-behaved simple Lie group, typical infinite dimensional representations of finite length are often not semisimple: those in the "principal series", etc. This is mirrored in a more elementary way in related Lie algebra representations, starting with Verma modules: these have finite composition series but are only rarely semisimple (then only when they are in fact simple).

The answers given by Bruce and Anton are sensible, but I'd add that it's important in Lie theory to distinguish the behavior of reductive groups from that of arbitrary Lie groups: finite dimensional representations of the former are semisimple (motivating the label "reductive") but often not in general. It's also important to distinguish finite and infinite dimensional representations, since you use the term "continuous". Even for a well-behaved simple Lie group, typical infinite dimensional representations of finite length are often not semisimple: those in the "principal series", etc. This is mirrored in a more elementary way in related Lie algebra representations, starting with Verma modules: these have finite composition series but are only rarely semisimple (then only when they are in fact simple).