group theory - What is an $mathrm{SU}(2)$ Triplet?

Under $\mathrm{SU}(2)$ group, a doublet transforms like: $$\phi \rightarrow \exp\left(i\frac{\sigma_i}{2}\theta_i\right)\phi.$$ The doublet looks like $$\binom{a}{b} ,$$ which is easy to understand because the matrix representation of $\mathrm{SU}(2)$ is a second order square matrix. What is a $\mathrm{SU}(2)$ triplet? What does it look like? And what's the matrix representation? Is it the $\mathrm{O}(3)$ group? If their is a $\mathrm{SU}(2)$ doublet, is there a $\mathrm{SU}(2)$ singlet?

I'm not very familiar with group theories, so I would much appreciate a detailed explanation. Also, what do these doublets and triplets mean physically?