There are several ways on which species can reproduce sexually with more than two sexes. These are the simplest (and there are way others).
Option a: X/Y system with YY sex (two individuals)
Human system of sexual chromosomes X and Y giving XX and XY (which is similar for other species like birds, which use ZW and ZZ) can be extended to three sex determinations XX, XY and YY. In this system there would be three sexes, which all can engage in sexual intercourse with each other. I'll not describe sexual organs! This has two suboptions:
Option a1
Each individual can have sexual intercourse with every other individual, including same-sex. This implies there is no definite "female" sex that lays eggs or can be pregnant.
- XX - XX gives only XX descendence
- XX - XY gives 50% XX descendence and 50% XY descendence
- XX - YY gives only XY descendence
- XY - XY gives 25% XX descendence, 50% XY descendence, 25% YY
descendence
- XY - YY gives 50% XY descendence, 50% YY descendence
- YY - YY gives only YY descendence
Idea can be that eggs with haploid configurations mix with each other in water like frog eggs do with frog sperm. Also something more elaborate with internal fecundation (which I again do not describe).
Option a2
Each individual can have sexual intercourse only with different sex individuals. This yields the same table as above, but with three options removed:
- XX - XY gives 50% XX descendence and 50% XY descendence
- XX - YY gives only XY descendence
- XY - YY gives 50% XY descendence, 50% YY descendence
Ideas for fertilization can be more elaborate, especially for internal one, but still there is no "female" sex.
Option b: X/Y/Z system (two individuals)
Human system of X and Y giving XX and XY can be extended to a third chromosome type Z, giving (in principle) options XX, XY, XZ, YY, YZ, ZZ. Then these need to be assigned to three sexes, or rule that some of them are impossible (e.g. YY is impossible for humans). There are so many subtions that I will not explore them all. Some are:
Option b1
XX is female, XY and XZ are two different types of male. Females have sexual intercourse with males of the two types (I do not rule out homosexual relations, it is simply that they are not interesting for breeding). Each relation can give offspring of female or same male type of the father.
Option b2
XX, YY, and ZZ are three types of female, and XY, XZ and YZ are three types of male. All types of male can have sexual intercourse with all types of female. Offspring have the same chromosome as the mother and one of those from the father, which means that it may be different from them both (e.g. XX female and YZ male yield an XY male or XZ male).
Option c: Triploid system (three individuals)
In these systems the chromosomes come in triplets, not in pairs. This can happen with only one type of sexual chromosome, X, which can be present or not in the gametes, giving XXX for females, XX0 for "half-females" and X00 for males (000 being impossible). Or it can happen with two types of sexual chromosomes, X and Y, for which options XXX, XXY, XYY and YYY must be assigned to different sexes (or ruled impossible). Or even more complicated settings.
Please read this answer in conjuntion with that of Monty Wild. His gives an evolutionary view about how this could have happened.