old maid: a single woman regarded as too old for marriage. Source
spinster: a woman who is not married, especially a woman who is no longer young and seems unlikely ever to marry. Source
These 2 above words refer to female, but now I want to find the similar term for male.
You can say that we can use "unmarried man" which is similar to "unmarried woman"
However, unmarried woman is "a woman who is not married" Source and it does not have the connotation as of "old maid" or "spinster".
It is because that "unmarried woman" could be the sister in a church who just do not want to marry because her religion. On the contrary, "old maid" or "spinster" means she is not attractive enough to get a husband or noone wants to marry her.