Identifying a painting by an artist can be easy if the artist's signature is clearly visible. Things become more complicated, however, when no signature can readily be seen, or if the authenticity of a painting is in doubt. In order to properly identify the artist, you need a little knowledge and a willingness to do some detective work. Depending on how intensely you pursue the question, it can be as easy as checking the local library or as difficult as solving a centuries-old crime.
Study art history. Know the different historical movements, the masters who shaped them and the ways in which they influenced subsequent artists. Understand the methodologies used in different artistic periods as well: the brushes, the paints, etc. Knowing the style of a given painting can help you identify the movement during which it was made, and knowing the materials used can further narrow down the date it was painted, both of which go a long way towards determining the artist.
Research as many different artists as you can. Every artist has a unique style, shaped to a certain extent by the period in which he lived but also reflecting his one-of-a-kind outlook. Intimate knowledge of that style can help you spot a particular artist--with a quick look, in some cases. You may notice little details and flourishes that pinpoint the creator of a work. If you suspect that a given piece was created by a specific artist, you can research that artist in particular, but a broad knowledge of many different artists can steer you in the right direction even if you have no idea who painted a particular work.
Check the painting closely to identify its characteristics. Start with the style itself: the techniques used to create the image, the subject matter. If you have access to the right materials, test the painting for its age and composition; contact a local university or museum for advice on the best way to approach it. Look for clues that point to a particular artist--a specific model or a particular subject--and extrapolate from there. The more evidence you can assemble pointing to a given artist, the more you can eliminate other possibilities and definitively determine a painter's creator.
In some cases, it may not be possible to isolate a specific artist. If the painter isn't well known, for instance, you may have to settle for an educated guess. Do your best when attempting to determine an artist's identity, but understand that there may always be a little doubt.