While the terms shown below have different formally-denotated definitions, they overlap considerably in terms of their connotations, that is, the emotions or meanings the terms imply...  

"ART"

An activity or product done by people with a communicative or aesthetic purpose—something that expresses an idea, an emotion or, more generally, a world view.  

• "Artist"

A person engaged in an activity related to creating art, practicing the arts, or demonstrating an art.  

• "Artisan"  (n.); "artisanal" (adj.)

A person skilled in making a product by hand; a skilled worker or craftsman; a worker in a skilled trade; a skilled manual worker who uses tools and machinery in a particular craft; a person who displays great dexterity.

• "Fine Art"

A visual art considered to have been created primarily for aesthetic and intellectual purposes, often by formally- trained makers, and judged for its beauty and meaningfulness; specifically, painting, sculpture, drawing, watercolor, graphics, and architecture.

• "Folk Art"

Visual art made in the context of folk culture. Sometimes this is termed "naïve art," meaning visual art created by a person who lacks the formal education and training of professional or studio artists.  Another common term is "outsider art," indicating its creation by self-taught or naïve art makers. Typically, those labeled as outsider artists have little or no contact with the mainstream art world or art institutions. Definitions of "folk art" vary, but often the objects have practical utility of some kind, rather than being exclusively decorative.  

"Folk Art" is very much a catchall term, often simplistically equated to "folk crafts," but used particularly when aesthetics are emphasized by the creator or viewer.  Taking the broadest possible view: the Museum of International Folk Art claims that this term encompasses art that "may be decorative or utilitarian, may be used every day or reserved for high ceremonies, is handmade, may include handmade elements, as well as new, synthetic, or recycled components, may be made for use within a community of practice or may be produced for sale as a form of income and empowerment, may be learned formally or informally or be self-taught, may include intangible forms of expressive culture like dance, song, poetry, and foodways, is traditional, reflects shared cultural aesthetics and social issues, and is of, by, and for the people; all people, inclusive of class, status, culture, community, ethnicity, gender, and religion. Moreover, since traditions are dynamic, traditional folk art may change over time and may include innovations in tradition."

See also the extended discussion "Concepts in Folk Art" in Wikipedia for a wide-ranging historical overview oriented mainly towards these terms and concepts in the United States. 

"CRAFT"

A pastime or profession that requires particular skills and knowledge of skilled work. Producers of "craft" may term their products as "art."  BUT: the term "craft" is often MISUSED in advertising as a marketing term for mass produced items.
See also: Craftwashing

• "Contemporary Crafts"

The diverse and eclectic artistic crafts of today, produced or conceived in the second half of the 20th century or in the 21st century. Contemporary artists work in a globally influenced, culturally diverse, and technologically advancing world. Their art is a dynamic combination of materials, methods, concepts, and subjects that continue the challenging of boundaries that was already well underway in the 20th century. 

• "Elite Crafts" ( = "High," "Academic," or "Fine")

This is not a usual term in the world of crafts or arts, but it is meant to designate formally-learned craft practices (also known as "studio crafts" or sometimes as "contemporary crafts").  This level of practice revolves around selectivity (i.e., juried and judged selection for exhibition) and formal training, as often established at colleges or universities.   

• "Folk Crafts"

Sometimes equated with "traditional crafts," or, broadly, with "visual arts" (since the products of craft production are viewable.) 

• "Handicrafts"

Artisanal handicrafts or "handmades," are any of a wide variety of types of work where useful and decorative objects are made completely by one's hand (hence the term handicraft) or by using only simple tools like scissors, carving implements, or hooks.  Some crafters refer to this category as "handmade arts and crafts." Synonyms for "handmade" may include: "handcrafted," "handwrought," "hand-crafted," "handicraft," "homemade," "homespun," "homebrew," "homegrown," "self-made," or "do-it-yourself" — though each of these carries a different and unique connotation.   See also: "Handicraft" (Wikipedia)

• "Heritage Crafts"

"Practices which employ manual dexterity and skill, and an understanding of  traditional materials, designs and techniques to make or repair useful things."   

                                  See: Towards a Definition of Heritage Crafts by Hillary Jennings

This term is particularly important in the United Kingdom, where the amazing Heritage Crafts Association (a UNESCO accredited NGO for Intangible Cultural Heritage) works to advocate for traditional heritage crafts. In partnership with Government and key agencies, Heritage Crafts provides a focus for craftspeople, groups, societies and guilds, as well as individuals who care about the loss of traditional crafts skills, and works towards a healthy and sustainable framework for the future.  

• "Maker" The Oxford Dictionary describes Maker as "A person or thing that makes or produces something."
  In the twentieth century the term Maker has been increasingly attached to a person who makes or produces something that is handmade.   
  • "Makerspace" 

• "Popular Crafts" 
  Craft objects generally recognized by members of a society as dominant or prevalent at a given point in time. Popular culture — including the concept of "crafts" — also encompasses the activities and feelings produced as a result of interaction with these dominant objects. The primary driving force behind popular culture is mass appeal; and knowledge of "popular crafts" is primarily learned and transmitted through mass media, including print, broadcast, television, and the internet.  

• "Traditional Crafts"

Traditional craft tends to generate craft objects out of necessity or for ceremonial use, often with reference to an expressive tradition based in a folk group, locale, or community. 

See also: UNESCO statement on "traditional craftsmanship." 

• "Studio Crafts"

Studio craft is the practice of craft methodology in an artist's studio or atelier. Studio craft produces craft objects at the whim of the maker or intended owner. These objects are often only desirable for use, and sometimes are more "artistic" or "sculptural" in intent, in which the crafter has downplayed or ignored the resulting object's "utility." 

• "Rural Crafts"
  Rural crafts are the products of traditional crafts production that is carried on, simply for everyday practical use, in the agricultural countryside. Once widespread and commonplace, the survival of some rural crafts is threatened. Rural crafts are not typically considered part of arts and crafts, as they are produced for a practical means, and not for leisure.  –Wikipedia

"Juried" vs. "Non-Juried" Craft Shows, organizations, festivals, and fairs:

        When a show or sales outlet, or membership in a craft organization is “juried,” it means that craftspeople must apply to participate, and that the quality of products as well as crafters' level of experience will be taken into consideration by a panel of judges. Typically, would-be participants are required send in a resume, photos/samples of their products, and a jurying fee. The jury is looking to determine whether or not the quality and marketability of products align with those of the craft fair. They will also make sure that a variety of products is being offered, so that each seller has a good chance of making sales.

      "Non-juried" craft shows, and craft sales venues typically give out spots on a first-come-first-serve basis, usually charging a "set-up" or "booth rental" fee to exhibitors.