A classification of primitive elements in a free group of rank greater than two is a hard problem, and there is no really satisfactory classification known. I am pretty sure [this paper of Shpilrain][1] is pretty close to the last word. As for elements representing simple closed curves, this is also not easy, and the best results are algorithmic results of D. Chillingworth (MR0248819 (40 #2069) 
Chillingworth, D. R. J.
Simple closed curves on surfaces. 
Bull. London Math. Soc. 1 1969 310–314), which were essentially replicated by Birman-Series (MR0744104 (85m:57002) 
Birman, Joan S.(1-CLMB); Series, Caroline(4-WARW)
An algorithm for simple curves on surfaces. 
J. London Math. Soc. (2) 29 (1984), no. 2, 331–342. ) and Cohen-Lustig (MR0895629 (88m:57016) 
Cohen, Marshall(1-CRNL); Lustig, Martin(1-MIT)
Paths of geodesics and geometric intersection numbers. I. Combinatorial group theory and topology (Alta, Utah, 1984), 479–500, 
Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987. 
57N05 (11F06) )



  [1]: http://www.sci.ccny.cuny.edu/~shpil/countprim.ps