1. Al Masjid Al Aqsa
The name “Al Masjid Al Aqsa” refers to the entire sanctuary (144,000 sq metres) and all the buildings in it.
It includes the famous grey domed building (Masjid Qibli), and the gold domed building (the Dome of the Rock), as well as many other buildings within the sanctuary (including Masjid Marwani, Masjid Buraq etc).
The picture below shows the boundaries of Al Masjid Al Aqsa.
Unfortunately in modern times, many people have mistakenly thought that Al Masjid Al Aqsa is only the silver dome which is located at the southern end of the sanctuary. However this is incorrect. The silver domed building has two specific names: Masjid Qibli (signifying the direction of Qibla), and Jamia Al Aqsa (signifying that it is the place that people gather for prayer). It is preferable to use these names when referring to the silver domed building to avoid any confusion.
To summarise the above, Al Masjid Al Aqsa is actually the entire area of the sanctuary, even though in modern times the name has unfortunately been mistakenly used to refer to just the silver domed building.
The entire sanctuary of Al Masjid Al Aqsa is also known by two other Arabic names: “Bayt Al Maqdis” (House of the Holy) and “Al Haram Al Sharif” (the Noble Sanctuary).
2. The Dome of the Rock
The Dome of the Rock (which is also known by its Arabic names Masjid As-Sakhrah and Qubbat As-Sakhrah) is the most well-known building in the sanctuary of Al Masjid Al Aqsa due to its iconic design and gold dome. It is situated in the middle of the plateau of Al Masjid Al Aqsa, and was built in 72AH by Abd Al Malik ibn Marwaan. It is a part of Al Masjid Al Aqsa and is not separate or distinct to it.
Summary
Al Masjid Al Aqsa is the entire area that comprises both what is today called Al Masjid Al Aqsa (the silver domed building at the southern end of the sanctuary), the Dome of the Rock (in the middle of the sanctuary), and everything else that is on the plateau (including Masjid Burak, Masjid Marwani etc) and within the walls of the sanctuary. The size of the sanctuary is 144,000 square metres.
The pictures below help to explain this clearly:
The picture above is also very helpful. Everything within the green borders is Al Masjid Al Aqsa, and the picture also highlights some of the structures and buildings that form part of Al Masjid Al Aqsa (inc. Masjid Qibli, Dome of the Rock, Masjid Marwani and the wall of Masjid Buraq etc).