Determine if your work of art is an original or a reproduction. How this is done varies greatly depending on what kind of art it is.
Determine what artist created the work. Look for a signature on a painting, or a signature or other identifying mark on a sculpture or work such as pottery. The identity of the artist will make a large difference in the value of the work, particularly if the artist is well known.
Learn the provenance of the piece. The provenance is the history of when and where the piece was made, who has owned it since that time, and the galleries, museums or shows at which it has been displayed. Particularly in the case of high end artworks, a work without a verified provenance will be viewed with suspicion by potential buyers.
Take your work of art to an appraiser who is recognized as an expert in that type of art. The world of art is vast, and no single person can understand it all. An appraiser may be a world authority on Impressionist paintings, but if you bring a piece of Minoan pottery to be appraised, he won't necessarily know any more than you do about it. Do your research before approaching an appraiser, learning about his area of expertise and professional reputation.
Determine your own reasons for wanting to know the value of the art before getting it appraised. This will affect what type of appraisal you get. Two basic types of appraisal are the "Fair Market Value" appraisal and the "Replacement Value" appraisal. The former is generally used for tax purposes when artwork is being donated to a museum or institution, while the latter is used for insurance purposes.
Ask the appraiser to do a search of the artist's other work, particularly what other work by the same artist has sold for in the last three or four years. With major fluctuations in the art market and the economy, price records from farther back than this often don't have a lot of relevance to current value.