Trinary or base 3 is the number base where numbers are represented using only 0, 1 and 2. Here is a quick base 10 to base 3 conversion:
1. 0001
2. 0002
3. 0010
4. 0011
5. 0012
6. 0020
7. 0021
8. 0022
9. 0100
10. 0101
11. 0102
12. 0110
13. 0111
14. 0112
15. 0120
16. 0121
17. 0122
18. 0200

That is of course using regular trinary as you can simply figure it out with a bit of basic mathematical knowledge.

Balanced Trinary

A nicer way of using trinary (to its full potential!) is a method of writing ternary instead of using 0,1,2 but to use -1,0,1. This is quite handy for use in programming with trinary (Malbolge comes to mind.. *shudder*), well in theory anyway.

NB: Some people call this ternary too.

Contrary to the belief of some, trinary is a nonword invented by uninformed computer scientists to rhyme with binary. Ternary describes "three" as a cardinal number (e.g. number of objects or subdivisions, as well as 3-valued number systems by extension), whereas tertiary is the successor of secondary.

There are thus three terms competing for the positions of numerical base, cardinal adjective, and ordinal adjective, only two of which happen to be valid, according to the OED: ternary, trinary, and tertiary.