I have seen the three words frequently used to express similar meanings. But what exactly is their difference that tells me which is the most suitable one in a certain case?
In terms of business/economic discussion (especially in investing) sector is a broader term than industry. According to Investopedia [emphasis added]:
Sectors describe broad categories within the overall economy:
However, this is not a hard and fast rule especially when the words are used independently, in which case they are often synonymous. Likewise, the term "industry sector" can also be used to refer to broader sectors.
Industry is often used (in the United States) as a formal means of classifying occupations and employment. The Bureau of Labor Statistics (BLS) defines industry as:
While "industry" and "sector" are often used in discussing economics, "field" is a broader term and not necessarily related to any sort of enterprise. The OED for American English gives the definition for field as:
For the most part, they are used in different contexts.
None of these distinctions is hard-and-fast. An insurance agent, for instance, may claim professional status by describing himself as working in the field of personal protection. Industry may come in handy in a bio or resume to describe an individual's varied career: “I have held responsible positions in every aspect of the housing industry”. Sector may replace industry when speaking narrowly of a particular aspect in which different domains are compared: for instance, an analyst may contrast the performance of the construction, manufacturing and service sectors on the stockmarket or in creating jobs.