I would like to use the NEO-FFI instrument to measure the Big 5 psychological traits. Is a license needed? Are there any open, free, or public domain Big 5 tests that can be used for no (or minimal) cost?
Yes, a license is needed for most uses of the NEO-FFI. Here's a quote from PAR's Permissions & Licensing page:
Anyway, yes, there are plenty of free tests! One such resource my former advisor (Dan Ozer, personality assessor extraordinaire) has always seemed to favor is the International Personality Item Pool. It might give you a headache to see how many options you actually have just through this source, just for measuring broad personality traits akin to the Big Five. In our work, we've generally used the Big Five Inventory (BFI; John, Donahue, & Kentle, 1991), which is also freely available for non-commercial use, but is somewhat basic (44 items, 5 factors, no facets). If that's not basic enough for you, you may also want to consider Saucier's (1994) Mini-Markers, which has 40 items and one fewer Likert scale option. Even shorter is Rammstedt and John's (2007) 10-item short version of the BFI!
I should add that this question is a partial duplicate: @JeromyAnglim already asked (and answered!) the question about free alternatives here: What free scientific measures of Big 5 personality are available? His list is more comprehensive, and I'm about to go add Rammstedt and John's over there, so this post won't have anything unique from that thread once I do (except the bit about the NEO-FFI licensing).
One might almost be inclined to say there are too many options! As for me, I'm with Martha Stewart on this one: "It's a good thing."
John, O. P., Donahue, E. M., & Kentle, R. L. (1991). The big five inventory—versions 4a and 54. Berkeley: University of California, Berkeley, Institute of Personality and Social Research.
Rammstedt, B., & John, O. P. (2007). Measuring personality in one minute or less: A 10-item short version of the Big Five Inventory in English and German. Journal of Research in Personality, 41(1), 203–212. Available online, URL: http://www.westmont.edu/_academics/departments/psychology/documents/Rammstedt_and_John.pdf.
Saucier, G. (1994). Mini-markers: A brief version of Goldberg's unipolar Big-Five markers. Journal of Personality Assessment, 63(3), 506–516. Available online, URL: http://darkwing.uoregon.edu/~prsnlty/SAUCIER/Saucier.Minimarkers.Full.pdf.