An original approach for the construction of state observers for dynamic systems is presented in this article. New notions of observability (delay-observability and delay-drift-observability), useful for the development of the observer equations and for the convergence proof, are introduced and compared with other standard observability concepts. In particular, it is shown that delay-drift-observability is an extension of the well-known drift-observability property. The construction of the proposed observer is straightforward and does not require any coordinate transformation of the system into a canonical form, which is a common feature of many other approaches. A numerical example is reported, where a system that is not drift-observable is considered. In this case, the proposed observer is easily constructed and performs well.