Why is a singlet state called singlet and a triplet state called triplet?
The terms arose back in the early days of quantum physics when spectral lines that were expected to be singlets were actually observed to be more complex (doublets, triplets, etc.).  

An electron can have a spin quantum number of $+\frac12$ or $-\frac12$.  For a system that exists as a singlet, all spins are paired and the total spin for the system is $S=0$.  If we have a single electron, $S=\frac12$.  If we have a triplet system with 2 unpaired electrons, $S=1$.  Also associated with these electrons/ systems is a spin-angular momentum vector, $L$.  Quantum mechanics tells us that $L$ is allowed to have $2S+1$ distinct values.  So

for a singlet, $L = 2(0) + 1 = 1$     only one state exists (singlet)

for a doublet, $L = 2\left(\frac12\right) + 1 = 2$   two states exist (doublet)

for a triplet, $L = 2(1) + 1 = 3$     three states exist (triplet)