Recessive Trait
A recessive trait appears only when an organism has two copies of the recessive allele, one inherited from each parent.
Definition
A recessive trait appears only when an organism is homozygous recessive, carrying two copies of the recessive allele (for example $pp$), because a single dominant allele in a heterozygote ($Pp$, a carrier) is enough to mask it. That means an organism actually showing a recessive trait tells you its genotype for certain, unlike a dominant trait, and two carrier parents who both display the dominant trait can still have offspring with the recessive one. Mechanistically, a recessive allele usually encodes a nonfunctional or absent product, and a single functional copy from the other allele suffices for normal function, so recessive disease alleles can persist at real frequencies because unaffected carriers are not removed by selection; by Hardy-Weinberg, if the allele frequency is $q$, the carrier frequency is $2pq$ and the affected frequency is $q^2$, and X-linked recessive traits appear more often in males, who carry only one X chromosome.
Example
Crossing two carriers ($Pp$) for white pea flowers gives $PP$, $Pp$, $Pp$, and $pp$ offspring, so $1$ in $4$ shows the recessive white trait while $2$ of the remaining $3$ purple offspring are carriers. Cystic fibrosis affects about $1$ in $2{,}500$ people of northern European ancestry, so $q^2 = 1/2500$, $q = 0.02$, and the carrier frequency $2pq \approx 0.04$, roughly $1$ in $25$, while red-green color blindness, being X-linked recessive, affects about $8$ percent of males but well under $1$ percent of females.
Key Insight
Most recessive alleles hide in carriers rather than in affected individuals, and for a rare recessive disease carriers outnumber affected people by roughly $2/q$, which for cystic fibrosis works out to about $100$ to $1$.