Disclaimer The opinions expressed herein are my own personal opinions and do not represent my employer's view in any way.
Conceptual Design
A conceptual design is an abstract or high level design which includes only the most important components and entities. The main goal of a conceptual design is to provide an understandable picture of the overall purpose of the proposed solution. Components may include major technology systems, external systems that are required for integration or overall functionality, high level data flow, and system functionality. Think of this as the "black box" diagram where portions of the diagram may be simply a technology component to-be-named-later but is identified with its role and purpose.
Logical Design
A logical design is a more detailed design which includes all major components and entities plus their relationships. The data flows and connections are detailed in this stage. The target audience is typically developers or other systems architects. However, it is possible to create logical designs for business purposes to ensure that all components and functionality is accounted and well understood. Logical designs do not include physical server names or addresses. They do include any business services, application names and details, and other relevant information for development purposes.
Physical Design
A physical design has all major components and entities identified within specific physical servers and locations or specific software services, objects, or solutions. Include all known details such as operating systems, version numbers, and even patches that are relevant. Any physical constraints or limitations should also be identified within the server components, data flows, or connections. This design usually precludes or may be included and extended by the final implementation team into an implementation design.