What is the difference between nouns "size" and "dimension". When I want to set dimensions/sizes of an object which noun should I use?
You can find both words in most of dictionaries. As a physicist I would not use the word "dimension" in a daily life. For me, this is a mathematical term designated to a property of space. The noun "size" refers to geometrical properties of objects that surrounds us. Below are some daily meanings of the noun "dimension" from dictionary.com.
a. measurement in length, width, and thickness.
b. scope; importance: the dimensions of a problem.
Usual for daily life, such as cloth size (L,M,S), freezer size (storage volume in litres), cup size (Large, medium)
Usual for engineering detail, such as dimension of a rectangle shoe box (length x width x height)
Dimension means size for engineering, formally or informally, and has a vector property associated with it. One can refer to the dimension "length" of a couch when buying furniture or on a mechanical drawing designing it. "Size" is more general. The words are synonyms; it is the usage which differs. I must disagree with our physicist friend regarding the use of the word dimension in ordinary speech. People use it all the time when they are specifying the length property of day-to-day objects, especially how they will fit into a given space.